I don't know what your subject line:
Are all batteries with more ah higher than those with less ah?
is asking, but:
If this is the same vehicle as this thread:
https://endless-sphere.com/forums/viewtopic.php?f=14&t=115270
then I would recommend adding external batteries in a weather-resistant case (like Pelican, etc) mounted under the seat, etc. to get extra range without recharging.
Alternately, you can replace with a more efficient chemistry than lead-acid, which will give you at best half of the rated capacity. There are a number of 12v-replacement Lithium chemistry batteries that are likely to give you better range *and* a longer lifespan, as long as they can handle the current draw (amps, A), and as long as you can find one the same size as your existing lead-acid battery.
It's possible that the orignal battery actually was one of these, and that is why you got twice the range you now get (because even if rated for the same Ah, (amp-hours), the lithium battery would give you about twice the actual delivered capacity than the same-rated lead-acid battery. They'll also usually be noticeably lighter than the same lead battery.
The problem right now is figuring out what battery you actually need.
The system you have claims to be 36v, but it cannot be, because you only have one 12v battery according to the other thread, and if it were a 36v system, it would not operate on only one; it's controller would have a safety shutdown to prevent destroying the battery at about 30v or so (since a 12v lead acid battery is "dead" at about 10v).
But if it's not already a 36v system, you can't just plug in a 36v battery; that could damage the electronics.
Another problem is that a 36v system will draw only about 1/3 the amps (current) to make the same power that a 12v system does, so if yours is 12v, it's three times as hard on the battery as it would be if it was 36v. So if it is a 350W system, then if it were 36v, it would only draw about 10A or so from the battery. But a 12v system draws 30A to do the same thing, and that's a lot harder on teh battery, so the battery can't deliver the same total range.
So you first need to find out whether your system is really 12v, or 36v.
The simplest is to look at your charger. If it has a label, you can take a picture of it and attach that picture to your post so we can see it, or you can type out everything it says on the label *exactly* as it says it. Uusally a 12v system will have a charger with a max voltage of around 13.6-14.4v.
Alternately, if you have only one battery in the entire system, and there are none hidden in other compartments, and you replaced it with a 12v battery, and the system still operated normally, at the same speed as before, with the same readings on the battery meter under the same ride conditions, then it is a 12v system.
If you have a voltmeter or multimeter you can set to 200VDC, you can verify the system voltage by finding the thickest wires that go into the controller box that runs the motor. (usually where most of the wires go into a metal or plastic box). If it's a 12v system it's probably a brushed motor, and so will have two thick wires going to the motor, and two thick wires going to the battery, and a few thin wires going elsewhere. You'd measure the voltage at each of the pairs of thick wires, with it turned on but not riding it, and see if you read just 12v or not.
Once you know if it's 12v or 36v, then you can find a battery that does what you need it to. A 12v battery has to be able to supply at least 30A without much voltage sag (meaning, how much the voltage drops while the system is moving you at the max power level), to provide the 350W your system is rated at. (assuming it actually only draws that much; you'd have ot use a wattmeter to verify this if you want to know for sure).
