I have done this before:
The socket they leave behind has threads, usually standard M3 screws (the finer thread standard PC screw)
Order a new cooler for LGA 115x that uses a hard mount. Something like the Deepcool Gammaxx 400 (shown below) is a good option if there is enough room. But basically any cooler that just clamps down on the four corners. Typically they have you run a backplate with threaded posts and cooler is secured with nuts/thumbscrews.
Go to a hardware store, get some long m3 screws and several sizes of washer.
Install the cooler like this:
https://i.imgur.com/8Wvewcv.jpg
So that is a Dell backplate, with some long shaft M3 screws holding down the cooler. I used large nuts as washers, since I didn't ave anything quite the right size.
Other coolers that should also work. Deepcool Gammaxx 300, Cryorig H7, Cryorig M9i, Cooler Master Hyper Evo 212.