Hi Kate. Welcome to Just Answer.
The push-in holes aren't really the way to go anyway, so I would definitely use the side screw terminals.
However, it is a Code violation to put two wires around one screw.
That being the case, you can pigtail those two wires. I would be happy to explain how that works, and make a quick sketch for you, if necessary.
Do you have any extra wire laying around?
OK, there is another way, though it's not my favorite.
How long are the two wires that need to be connected to the same point on the switch?
It's not an absolute must that you pick up some new wire. Safety wise, there is no difference.
Do you have wire strippers and a sharp knife?
Great. How about wire nuts?
You only need one. A yellow would work, if you have one.
OK, let me make you a quick sketch. Will you be standing by?
Very good. Starting on it now...
It will meet Code, no problems. I'm not sure I understand the second part of your last question. I don't think I've ever seen a box that's limited to #14 wire. Here's the drawing...
Yes. Sorry the drawing is a bit rough looking.
You are most welcome. Is there anything else I can do for you?