To verify without buying a new thermometer (although it never hurts to have a spare)....try switching sides with it. Put the unit on the warm side and the probe on the cool side...and see what kinds of readings you get.
If the room you have the tank in is around 70 degrees, then it IS likely to actually be that cool in there. UTH's are great for creating a basking spot, but they won't warm the entire enclosure...especially if it's a big, glass tank.
Try getting another dimmer switch for the lamp so you can leave it on all the time as well. (But you'll need to get a red or blue bulb so it doesn't bother the snake...or a ceramic heat emitter...which screws in like a bulb but doesn't put out any light, just heat.) Sounds like you definitely need some supplemental heat to get the overall ambient temps of the tank up closer to 80.
EDIT: One more thought about a second UTH for the "cool" side....I wouldn't. Again, it will just create a small warm spot, without actually adding any additional heat to the tank. You need something to radiate warmth and warm the whole tank up to around 80.