What kind of tremolo is it? Is it a floating tremolo? For a non-floater it will help to add the extra spring. If you definitely are not going to use the whammy, then even screw the spring claw in until the trem is tight to the body.

If it's a floater, I think you'll have to block it. If you look around the Guitar Tech forum, there is some advice on how to block the trem.

One thing that could also be part of the problem is if the strings are new and/or cheap. Maybe try a new set and make sure they are stretched. By the end of a session you should be staying in tune.

There is a third possibility, which is the tuners slipping.

There might be some other, less likely factors, too, such as a loose truss rod (and your neck relief is gradually increasing) or even temperature changes from times of day or one day to the next.

But the trem is probably the culprit. I pile all the springs I have onto mine except my Floyd (which is the only one I actually use). Why don't they sell more hardtails?!? :D