Best way to improve torque or hill climbing ability?

I was successfully able to overvolt my mini pocket scooter (as shown in my avatar). Great results of higher speed and much improved hill climbing ability by just changing out the controller, replacing the 24v system to 36 and new throttle.

I'd like to do the same thing to another vehicle. Looking to replace the controller and upgrade from 24v to 36v.

BATTERY SELECTION:
I'm still an idiot when it comes to electronics though. Higher amperage I understand equates to longer range. Any reason to go with a battery with fewer amps? Does amperage affect torque at all? In my new project, hill climbing is more important than it was before. I don't want to fry anything. Totally okay with burning out a motor every now and then but would like to keep my controller and batteries in good shape.

My choices are three:
A. 12v 15ah batteries
B. 12v 12ah batteries or
C. 12v 10ah batteries

CONTROLLER CHOICE:
Getting a bit paranoid about overheating and melting wires again. If I have no intention of going more than 36v 15ah on a 500w motor, would there be any benefit (resistance to melting wires) by purchasing a 48v controller over a 36v one? Or is that just stupid thinking? Sorry, like I mentioned, I don't know much about this stuff. Trying to learn. Doing a bunch of research, but responses from you experts would help tons. Any and all comments welcomed. Thanks much in advance for your time and input.

Amperage-hour of the battery would affect torque if the amperage demand is extremely high, because the battery would be in the bad efficient discharge rate, you would get voltage drop. But you need a really high amps demand. The real difference would be the voltage drop and heat on the battery.

This depends on battery chemistry and controller máx amps discharge capability.

For SLA / lead acid battery more than 1C of discharge for example 10A on a 10Ah sla battery start to have a considerable voltage drop (more than 10%)
For Lithium-ion battery, close to 2C discharging rate (depending of the chemistry variation) 20A on a 10Ah battery would have either a considerable drop (close to 10%)
For LiFePo4 battery you need to discharge to more than 3C rates (depending on chemistry variations) 30A on a 10Ah battery to start to have voltage drop (close to 5%)

For continuous discharging rates at full throttle on hills for examples. This values would fit on 1000W controllers and motors on >48V setups or more. For your 500W motor and controller the differences would be the range.

Also you really need to know if the battery has a BMS that would be limiting the max continuous discharge rates, so you really would feel any differences in torque power.
For the battery itself, the amps capability is usually really high, 3C to 10C in most of the cases

If hill climbing is the primary objective, the best solution is to leave the battery, motor, and controller alone, and instead change the gearing to give more torque at the drive wheel !
But,...somehow I suspect you also want speed as well ?

You bring up a very good point. I do have an option to change out a sprocket (and yes, a bit of me has the speed bug). Logic says torque. Fun part of my brain says speed.
Fewer teeth = more torque.
More teeth = more speed.

Correct? At least that's what I understand.

1. How do each of those affect the stress on the motor, battery, and controller? Swapping out the sprocket for a new size sounds great, but I'd like to know what I can expect as far as sacrifices.

2. Is there a way to calculate how much less speed I'll have when I improve torque with a new sprocket?

You bringing up the gearing is making me think of something else:

SPEED DEMON VS HILL CLIMBER:
Recently, I was lucky enough to come across a second Cruzin Cooler. This one has a 300w as opposed to the current 500w.
300w motor one will have 36v 12ah.
500w motor one will have 36v 15ah.

If I were to make one into a speed demon and the other one into an awesome hill climber, which motor should I use for which? 500w for hill climbing b/c hill climbing requires more power and less likely to burn out? or reserve that for speed and use the 300w since hill climbing doesn't require tons of speed? Not sure how the logic goes. Please enlighten me.

Thanks in advance for your input.

No, a steeper gearing does not always equal greater speed.

Your best result will be to increase voltage, and gear it lower by a slightly smaller % than you increased voltage. This will always give you greater hill climbing and acceleration, because you've increased torque at the wheel throughout the rpm range. It will also always give you higher speed, because not only have you increased max rpm of the wheel, but you've also increased power by the % increase in voltage.

Another benefit is that the lower gearing means your motor motor is running at lower stress, which typically enables you to run higher current limits, since the motor spends less time at high load.

Note that this all assumes the motor can handle the new higher rpm.

Thanks John for addressing my questions. You too Etrike! Thanks for looking through my blogs. Hope you liked what you saw. More for my record in case I forget the stuff down the line. I keep these toys for years and years so it helps to have it all written down somewhere.

UPDATE:
I overvolted the 500w last Saturday night and brought it for a test run up and down the block Sunday morning. Top speed surprisingly did not increase according to my Speedometer app. I ran it stock and it topped out at 13mph. No change after the overvolting as far as top speed is concerned.

Any reason for that? I was expecting a proportional jump or at least something close. 13mph to maybe 20mph as opposed to the manufacturer rated 23mph.

Torque and acceleration though is improved a ton. Just barely touching the throttle makes the thing jump. A bit freaky and in fear, I ordered a speed controller b/c handling this at 2mph at areas where I'm not really supposed to ride would be tough. My daughter being very hands-on (as this ice chest is actually used as my baby stroller) makes it tough to be in full control of the throttle sometimes.

PLANS FOR HILL CLIMBING:
You folks said to gear down as I improve my voltage. Latter is completed. Next step is to gear down. I was wrong up top. It's the REAR sprocket need to enlarge in size to get a low gear (like a bicycle). It's the FRONT end (crankset) that needs to decrease in size to do the same. Since the chain motor doesn't have a sprocket any smaller, the only thing to do is to get a massive rear sprocket.

I counted the teeth on the stock sprocket and it's got 55T. The aftermarket ones by coolermods have a choice of:

68T
80T

Having spoken to them on the phone and doing my own simple calculations, I'm leaning towards the 80T sprocket to maximize the upgrade in torque (even at the expense of speed). The 68T is not currently available. It's got about 30 days before the company will fabricate more of this exclusive product. I might wait, but haven't really finalized it yet. I was gungho on the 80T prior to my overvolting. I figured some increase in top speed can be sacrificed for torque. Since the top speed hasn't improved at all, I'm not quite as excited about the 80T.

a clutch

Not sure if that's a joke. I would have no idea how to do that and not really think it would be worth even trying. Should be good for now. At the most, I'll swap out the stock 55T rear sprocket for a 68T or 80T one at the cost of top end speed and acceleration. The recent overvolting should take care of the difference and not make it that bad.