How do I solder to a soldered overlay board?
That board looks to have fully plated thru holes with pads on both sides.  That means the top side isn't more special electrically than the bottom.  You can just as well solder wires from the top as from the bottom.

Also, if you solder parts incrementally as you install them, then only one part at a time has to be held in place when you flip the board over.  Parts with bendable leads, like thru-hole resistors, stay in place after you bend the leads a little bit.  After you solder, you clip off the excess lead anyway, to the bent lead won't short against anything.