I have had good luck using blue painters tape. It is designed to be sticky enough to adhere to things, but also to peel away leaving the surface it was attached to undamaged.
I take a small strip, tack it to the stock in the bottom margin area and to the printing board.
The stock stays down when I lift the screen; I peel the tape off of the margin area, and then when I put the next piece of stock down, I reset the tape in the next margin area. A piece of tape is usually good for twenty or so prints before I need to replace it.
But when I grow up I want a vacuum table too.
Spray tack: ruins the back of your paper
Tape: ruins the edges of your paper
Vacuum table: ruins your ability to make shit prints
Seriously though -- the first few times I printed on paper, it was using spray tack and/or painters tape on a t-shirt press. Not fun. Build a vacuum table, they're cheap.
My wife builds 3-4 Rabbits a month now. Keeps her in smokes and plants for the garden.
Table saw, drill, screws, glue, 1 sheet of 5/8 plywood, 1 small sheet of laminate, some hardware bits and pieces - the drilling takes about 2 hours, the rest about 2-3 hours over a day or two (you need to let the glue dry)
If an old granny can do it, you young fuckers should be able to.
Squeegeeville — Rabbit Portable Vacuum Table for posters, stickers, art prints, signs