First, the exact tools(thermometer, UTH) do not matter(Think McGiver) You can get the job done in any one of a thousand ways.
First, is the snake feeding? Where does it go after feeding? Normally, a snake will always choose cool when not feeding. Then if cool enough will choose heat after feeding.
But this depends on other conditions as well. Like hiding spots. There should be secure hiding spots on both the hot and cool sides. The reason is, most snakes do not feel comfortable out in the open. They there are other factors like humidity and dehydration. With to low of humidity and related dehydration, a snake will refuse to feed(at any temperature) Then choose the cool side.
Of note, In winter, the temps normally drop and the humidity is normally very low. So such sad cages are tanks with screen tops are DISASTERS. Even turtle people know to cover turtle tanks(turtles in water) in the winter and when expose to air conditioning. As the turtles will dehydrate(turtles in the water)
Why snake folks don't get that is WAY beyond me. Kingsnakes, the subject of this forum, are fossioral, or in the ground, or inside something or another(dead tree, rock crevices, etc) They only come up duing events of higher humidity. Or around water sources, ponds, creeks, streams, lakes, etc, WHICH have a higher humidity around them.
BETTER TOOLS
You mention a stick on thermometer, and a UTH, both are the bottom of the barrel. Cheap and not very good. Do not restrict yourself with poor equipment. There are many better ways to accomplish a better enviornment for your snake. Infra red heat guns are GREAT for understanding cage temps and you can shoot every spot in the cage in a minute. EVEN THE SNAKE. They give temps the snakes actually use. You know, what they are sitting on. You see, air temps are not what snakes understand. THEY ARE NOT IN THE AIR. Or they should not be in the air. hahahahahahahahahahahahaha
So start over, give more information. Offer details, like what the snake is doing, etc.
ALso understand if the temps are too low, the snake has no choice but to go dormait. IT will do so because the conditions do not support activity. Offer conditions that support activity and the snake will be active.
Yes, its winter, but I just fed bunch of babies and they all ate.
I apologise for being blunt. Its just the way I am. Wait, I wasn't always like this. I am now a product of this enviornment, hmmmmmmmmmm just like the snakes.(think some sort of demented humor) Cheers