Well, without looking at your guitar and assuming you have setup the guitar well, no dings, bumps, stuff that messes up tuning, the usual course of action would be
1) Change the nut (bone, tsuq or graphtech)
Cheap and makes a big different in tuning stability.
Check, if tuning don't stay. then go on.
2) Change the tuners (google, research, a lot of options)
3) Change the bridge saddles (k-t-s, graphtech)
Something like that.
But i bet you haven't let a good tech look at it so why not do that first?