advice needed

I'm new at this so please be gentle :?
I have an ebike I'm trying to get the way I want, it has a 250W front hub motor, supplied with a 36V 6Ah Lithium battery limited to 15mph but I want a little more- I'd hope for 20mph or more- but nothing too much more. I have a 20Ah LiFePO4 ordered, the only identification on the brushed hub looks like a Z with a partial circle from top left of the Z clockwise round to the bottom left of the Z. I don't know if the make can be identified from this to get more info- perhaps it can give more than 250W so won't need changing. The controller is a proprietary one ad seemingly can't be de-restricted. I'm happy to stick with 36V (at least for now) but I don't want a flying machine.
So- what is the best way to upgrade it?
New controller to start with? Would this allow more than the 15mph the current one is limited to? I'd like one that can be power only at times, as well as pedal assist (I have medical issues so this is necessary).
New hub motor? Would a brushless one be better than brushed if buying a new one?
Or both controller and motor at the same time?

All advice gratefully received.
(and any links to sites to help learn appreciated)

A new controller would be needed to get more out of the hub. But we need a bit more info to delve into what to get and where. Is it a gear motor? Or a nearly completely silent direct drive. If direct drive, the motor can most likely take at least 500 watts with no problem. That would be a 36v 20 amp controller, but to get the right one, we need to know how many wires come out of the hub. 2 wires, brushed hub controller. 3 wires, brushless pedal first, or sensorless controller. 3 big wires and 5 small wires, brushless start immediate controller.

The easiest thing to do though, and not so expensive, is just get a kit for a bigger motor. Hightekbikes.com sells the Aotema, Ebikes-ca has many good options, particularly the 9 continents, and ezee, and Ampedbikes has good deals on the 9 continents too. Most of the Ebay kits come with batteries, so they are not so cheap if you get batteries you don't need. I like my Aotema from hightekbikes just fine, and have a long review thread going. There are threads for the others in the reviews also. Free shipping was offered for ES members by hightekbikes at one point, I don't know if it still applies now.

Nothing wrong with having two bikes, but if you don't mind fooling around trying to get the right wires connected on a new controller, you should be able to get 20 mph out of your existing hub, at least for awhile anyway unless you start going for reall loooooong rides with the new battery. The easiest controllers to figure out the wire sequence for are the pedal first ones, which will run any brushless direct drive motor. Even if the motor has hall sensor wires, the pedal first controller will not need them.

By the way, how many characters are you today? Even a good day with only 3 personalities must be hard on your coworkers and the boss. Let us know when you are one that we can be hard on.

hopefully our friend didn't scare you off by offering too many options. i am often guilty of that myself. sometimes in our desire to help we can deluge a newcomer with too much information.

first, if you want to go faster we need to understand what is limiting your bike. if there is a speed sensor that limits the actual speed, it will just stop supplying power at that speed even if you are coasting down a bit of a hill. If you have a speedometer you can take the wheel off the ground and see how fast it spins, and i expect you will find that it can spin faster than the bike will actually travel. If this is not the case disabling the speed sensor will give you some extra mph. I expect that your bike does not have a 15 mph speed limiter and that is just the max speed the controller will provide enough power to achieve. It is quite likely that if you provide more current to the motor you will go faster. It is also likely that more voltage will take you faster, partly because the controller can turn this into more motor current and partly because every motor has a specific relationship between voltage and rpm. hopefully this is not too confusing so far.

since you say you have medical issues i expect you may also have money issues, and that you would prefer not to spend any more than necessary to achieve the desired results on your bike. A 6 Ah battery is quite small even for a 250w bike, so it is possible the larger battery at the same voltage will provide more power to the controller just because the voltage will not sag as much under load. this assumes your new pack is the same voltage as the old one. If you have determined that there is not a speed limiter, see if the new battery lets you go fast enough before buying another controller. If you need a new controller there is a guy on ebay who sells perfectly good units for a very low price, once we determine what you need. okay up to now? if not just say so, we just want to help.

if your goal is still higher speed after you get the new battery, and the wheel will not spin at a little more than the desired top speed when off the ground, you will need more voltage. the only easy way to get that is with more batteries. it is however important to make sure your motor can handle the extra voltage. a better controller will not get you more speed if your motor cannot go faster at that voltage. If the motor spins at say 22 mph off the ground but you can only go 15 when on the road, a controller supplying more amps may get you there.

try to digest this, and tell us how many wires there are coming from your motor to the controller, and we will be better able to help.

let us know what kind it is because there are some upgrade kits for currie and the izip motors