The cheapest way would be to buy a sheet of plywood and cut it in half or to the size you need and then screw it to the current table. If you are interested in buying a new table Sam's and Costco have good prices on heavy duty plastic tables in various sizes. They have legs that fold flat so storage is easy.
The "plywood" idea is not bad, but you could probably find some thinner, lighter material than what I think of as "plywood" and cut it to a size convenient for the puzzle. Probably no need to screw it to the table, just sit it on top. That way, you can turn the plywood sheet around if you want to work on the other side, or pick up sheet, puzzle, and all and go into another area, etc. Or when you want to do something else on the table. Or work on the kitchen table, etc. Or stash the puzzle under the bead and work on it later.
We had a sheet of particle board and a piece of chalkboard material we used for puzzles and other projects when I was a kid.
I've seen some smooth, thin, fine grained particle board stuff that would be great for this kind of project. If you can find a piece of cardboard the right size, that would probably work well for this one puzzle, too.