Totally recommend a slight tighter girth. And don't be afraid to use a mount g block. Also look at the shape if your horses shoulders and back. Might be a need to have a shimable saddle pad. Also have you tried different stirrup lenghth. If you are naturally balled a certain weight will cause yourself will require adgustment in the stirrup leathers. Might just naturally be balanced more to get your self figures.
I had a riding instructor who told me that side to side slipping was because of the way I was sitting in the saddle, and I expect she was right in my case.
She'd check that the girth was tight enough before I mounted up, and like someone else suggested, she'd hold the other stirrup down while I mounted.
Most people have a weak side- as I surely did/do- and I'd notice the saddle slipping slightly after trotting. I suspect I was posting with more pressure on my right stirrup, so I tried to be conscious of that.
Like so many questions on here, the specifics of exactly you're dealing with aren't detailed, but if you happened to have the problem I had, I hope this is helpful.
Maybe try a breastplate? Does it slip side to side or forwards or backwards? If backwards then a breastplate will help, if forwards try a cropper (beware many horses buck with these so be careful) or do you girth straps on the front two straps rather than the 1st and last strap. If side to side, a saddlecloth or pad with gel grip will help or do you girth up tighter. Having a loose girth can be very dangerous so don't be afraid to do it up right as this is safer for both you and your horse, you may even after warming up do it up another hole or two as the horse deflates. Also maybe get someone's to hold the other stirrup and push weight down on it as you get on as this will prevent you saddle slipping early on.
Doing up the girth won't hurt him/her (horse) so don't be afraid to do that
How many saddle pads do you have underneath it? The no slips pads will absolutely help the saddle not slip but I find a long of horses hate the neoprene/rubber stuff on their skin. I imagine it is like putting a rubber band in your hair vs a hair tie and it pulls at the hairs. Plus that material is very HOT so you end up heating up your horses back more especially with a synthetic saddle because those are not wool flocked and are hot and hard on the back as well. I would get a nice wool pad and make sure you do not "pad up" and put more than one pad beneath the saddle. Having a girth with more elasticity to it will help as well. Try and put less weight in your stirrup when you are getting on and pull more with your left hand which should be pulling on the made NOT the saddle and pushing down on the back of the saddle rather than pulling. You twist your horses back less while mounting and the saddle should shift less. That is if you do not have a mounting block handy :)
I'd get a non slip saddle pad. I totally recommend an ecogold!!