My bp started out in a 20 gallon aquarium, and I used a UTH successfully. You definitely need to get it on a thermostat though, because they can get really hot and burn your snake. I use a hydrofarm thermostat but there are also more expensive reptile-specific ones. If you only have one thermometer on the cool side, that is not a safe way to be monitoring your temperatures. The UTH will not heat up your entire cage, it pretty much just makes a hot spot right above it. You need to get a thermometer with a probe so that you can put the probe above the UTH to see how hot that particular spot is. The accurite thermometers from Walmart work great for this, and they are not super expensive. You put the probe on the UTH side and the actual thermometer box on the cool side, and it will give you both of your temperatures. They will also tell you the humidity of your tank. When I had my 20 gallon I covered half of the screen lid with a wet wash cloth and that worked pretty well to keep the humidity up, along with misting if needed.
It's also a good idea to have 2 hides for your snake, one on the hot side and one on the cool side so that it can regulate it's temperature as needed.