I've had pretty good luck using the trem and staying in tune.
I use 5 springs, I set the spring tention so that the trem will have some upward bar action, in other words, I want pull up on the trem bar and go sharp with a note...instead of setting it like most folks do, where you can only dive bomb.
If you pull up with the trem bar, with the same force that you pushed down...it will stay in tune. If you are going to tremelo chords with the bar...try to picture "center" and base you trem bar movement on that being the center...go the same distance up as you go down.
Be aware that this entails loosening the screws that hold the trem to the guitar body...I wouldn't want someone to attempt this and strip the screw holes in the guitar out.
You have to loosen those screws to get enough pivit. The traditional strat set up is to loosen these screws until the tremplate lays flat on the guitar..to get upward bar movement you have to loosen them more than that.
This is supposed to be the way Hendrix set up his trem...it seems to be the only way to stay in tune and really dig in with the trem.
I've been playing a trem strat since 1976 and IMO a strat set up like this is the best you can do with the original hardware
Just make sure you have good keys also. I have Klusons. I use .10 to .48 strings (the heavier the string gauge, the better this works)