OK, but it's not shooting back at you, and going somewhere? Perhaps the sewer, or a landscape drain, or gutter/downspout drainage system. It'd be prudent to find out where it goes, or trust the original pool builder ran it somewhere that is OK.
So there was two ways to go about saving your overflow system (assuming you want to), and raising the level of it. But one of them required finding the other end. So down to one:
With some pliers (needle-nose maybe), grab a hold of the sheared-off scrap there and see if you can finish tearing it off. Gently!! You don't want to tear into any more of the skimmer or weaken the joint between the overflow opening and the overflow pipe. If gently prying it back and forth doesn't loosen it off, then switch to a fine hacksaw blade and cut it off, being carefully not to scratch up the skimmer surface. Apply duct tape to protect the skimmer PVC if you have to. You're just cleaning up the overflow opening.
Then find a piece of PVC pipe of an outside diameter that will fit inside the overflow pipe. It can't be loose fitting. You might have to file/sand it down to get it to fit, and fit snuggly. Cut it to length, so that it sticks out from the wall about 3/4" with the pipe snugged into the opening.
Now fit a 90° elbow onto the portion sticking out, and point it up. That will raise your overflow level about 3/4" of an inch. You can then cut a small piece of PVC and slide that into the upward-facing elbow, to extend the overflow level to any height you want. Just enough to keep water from flowing out, and below the level of the bottom of your coping. As water rises in your pool, it will eventually run into the PVC and away from your pool, before it overflows.
As Jim does with his plug, you remove the whole thing to get the basket out.
I would glue only one joint, where the section coming out of the overflow opening goes into the 90° elbow, and nowhere else. That'll give you a good "handle" to pull the thing out when you need to. You want the part going into the opening to be able to come out of the skimmer, so no glue there. And if you don't glue the short vertical piece of pipe into the elbow, you can easily swap it for one of a different length, should you later want to adjust the overflow level.
If you can't find a diameter of PVC to fit the opening, come on back and I'll walk you through another way to go about this.