I tried doing a search but all the results are about the jack itself being loose, which is not my problem. The issue is the cable slides in and out of the jack without "clicking" in tight. Is there something inside that maybe needs to be stretched out to give it some resistance?
Yes, there's a tab on the inside of the jack that holds the cable in by tension. Bend it back into shape a bit, tighten up the jack and nut just in case, and try again.

If that doesn't work, you probably need a new jack. Sometimes they can just disintegrate to the point of falling apart. The good news is that jacks are cheap and easy to come by (1/4" mono jack, panel mount, $5 or less) and easy to replace. If you don't know how to solder, this is a great way to learn. Soldering irons are cheap, too (30-50 watts, fine cone or chisel tip, not a "gun" or plumber's iron; fine rosin-core solder, not acid core).
Roc is right on -

Unscrew the whole jack plate and insert the plug into the jack with it out and watch how it goes in. It should become obvious how the mechanism is supposed to work: there's a prong/s with a little bend in em to match the head of the jack; if that is bent out, it won't hold the jack in! Not all jacks/plugs are made equal as well unfortunately (at least in my experience) and may not be lining up at all. If you can't get it right, get a new jack.

Roc is correct that both jacks and irons are cheap. If you don't know how to solder, don't be intimidated by it, watch a few videos on youtube and you'll be good to go! Pay attention to where the wires are on the old jack, I'd take a pic. Match em up with the new one, voila!

Once more, listen to Roc about the rosin-core, and a simple iron, nothing huge.

I like to take Elmers wood glue and fill the entire output jack cavity in a cement-like and oddly mucus-colored block. Before it sets, I put the cord in and let the glue dry all around it. Never have to worry about it again. Also make sure your guitar case allows for room for the cable to stick out of the guitar... and also the rest of the cable.

Added benefits: If you drop your guitar you can always reel it back in!