Off the hill, cruising in no bad wind, a given speed will take a given wattage. For example, 400w gets most bikes about 20 mph.
So with a 48v 10 amps controller, 400w is about 9 amps. With a 48v 100 amps controller, 400w is,,,,,, about 9 amps. So there is no penalty in range for having a big controller you aren't using any more amps from. No benefit either. If you start up faster, then it costs you, but it could be well worth it.
On the hill, all anybody can really say is it depends. It depends on the motor winding, the rim size, the weight of the rider, the grade of the hill, the voltage you are running, if the battery can stand it, and the type and or size of motor. Oh so fun running any kind of valid experiment with 8 variables besides the 9th variable, controller amps.
So you have to go test it with you, on your bike and motor, on your hill, with your battery. But some generalizations emerge. IF you are now stalling the shit out of that motor, more amps will help your motor maintain speed up that hill. In that case, faster may well be less power, because you were previously running at about 10% efficiency. Get efficiency back up to 60%, and it helps.
If you have a typical direct drive hubmotor, and you get up that hill at 15 mph, you are doing ok. You don't need more amps. If you are slowing to below 10 mph, you are about to melt that motor. The solution if you have wicked hills may be to get both a bigger controller, and to run a slower winding motor. If you want to keep the top speed the same, then that means more volts to get to that same speed with the slow motor.
Or, get really huge motor, and heave huge watts at it, and fly up the hill. Works great, but not particularly good for great range.
But I digress. You said mild hills. Again like on the flat, if getting up that hill is only taking you say 18 amps, then the bigger controller will get you up the hill using ,,,,,, wait for it,,,,,, 18 amps. If the hill was taking you 22 amps, the typical max real world amps of 20 amps controllers, then a bigger controller might get you going up the hills at 27 amps. Then we get back to , does the motor run more efficient at 27 amps, at that grade, with you on that bike? It could well be that climbing the hill faster at 27 amps is better.
So if you know what amps you are using going up those mild hills, you can better decide. If you are not maxing amps on those hills, a big controller will not help you much. A given motor on a given grade with a given weight rider may never pull more amps than you already use, unless you change other variables like the winding or the voltage. If you only use 18 amps now, a 35 amps controller won't go faster up that hill.
The best way to ride and get more range is to use a wattmeter. Then you can learn where your watts are wasted, and where they aren't. Changing your riding style can make huge gains in range, even if you go about the same speed as before.