Welcome to the forum and the wonderful world of snake keeping!!! I have a Vanilla BP named Tyson....
Yes, DEFINITELY get a thermostat. It's probably the most important tool for snake keepers. Unregulated heat sources can spike to dangerous temps and can cause serious burns or worse to your snake. Google search snake burns real quick and you'll be out buying a t-stat before the pictures are done liading...
Also, you want to measure the temp on the surface of the enclosure. Snake can and WILL burrow and find the hottest spot eventually, so that's what you measure. You will need a temp gun for that.
You're also gonna want to ditch that reptile carpet. They are horrible for snakes and just harbor mold and bacteria.
Just to clear things up, your Heat mat (UTH) is there solely to provide a hot spot and not for ambient temps. Your light bulbs are there to provide ambient temps and a temp gradient. BOTH should be regulated by a thermostat.
Feel free to ask any more questions, we're happy to help
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