If you hold it in the right grip first and pull back on the plunger as soon as you break through the skin you'll find that your hand will already be in the correct position. By this I mean you actually set up your hand positioning in a way that means you dont actually need to shift much if at all. It takes a bit to get used to this but I can almost guarantee you're probably doing it the noob way which is you're just holding the needle for breaking the skin, going until you hit a vein, then fiddling around with the plunger. It's why many people miss.
I can hit myself 95% of the time, first time I go even though my hands shake a fair bit. The area I lack skill in is choosing the correct angle more than anything else but I've also got some luck in that when I get the right angle, I actually have to insert the needle basically all the way to the barrel to hit the vein by keeping it pretty flat so there's not really much ability for it to slip out.
Practice with some saline if you can, just inject that for a bit after you mess around figuring out which position you should start with. It's really important to be pulling back as you go in, I don't know how so many people don't know about this aspect of injecting but the last couple of people I've used around just seem to go fishing. Pulling back means that as soon as you hit the vein, you'll automatically register and if your hand is already in the correct position for pushing then you don't need to adjust the grip, you can just go.
It is admittedly really awkward hand positioning initially but trust me you want to adapt to it. Doing it the other way just results in sticking yourself and slipping out more often than not.