For a shower stall drain, Oxo has a product that you might like. It’s stainless steel and just sits over the drain. There is a silicone lip to keep it from sliding around. For my bathtub drain, I use one of the type that ThelmaLou linked to. You do have to remove the pop up drain plug to use it though. But I never take baths, so I don’t care.
Possibly you could wear a shower cap while in the shower, and wash your hair in the sink, (if the sink does not get clogged). In addition, you could cut your hair, therefore less hair would go down the drain.
I would have just said use a plastic strainer as others suggested already, but I did see someone who stuck a copper pot scrubber into their old style sunken bathtub drain. Hardly worth refitting an existing tub for, but you can take the metal strainer off and do that.
The problem with a solution like that is that as hairs get caught, they will reduce the flow of water, and they are going to be difficult to clear from the pot scrubber. A purpose-made strainer is better.
I didn’t ask about it, but I agree, I think you’d have to throw those away pretty often.
I would unscrew the drain cover, cut a piece of screen the same size, and screw the cover back on with the screen piece underneath it.
It would get plugged up with hair, and be difficult to clean it off.
As I noted above, it may be necessary to remove the collected hair/lint/whatever from the strainer several times during one shower. This may be seen as a nuisance, but one must simply deal with it, as a normal part of shower-drain-strainer operation.
Also, if you go the fabric screen route: Buy several square feet at the hardware store. The little piece you cut for your drain will gradually get frayed and frazzled over time, and probably needs to be replaced once a year or so. It’s trivially easy to cut a new piece, and a few square feet of fabric screen should last you for years and years.
ETA: Just a tangential thought, since you talk about unscrewing the drain grille. It’s a good idea to do this once in a while anyway, just so it won’t get so corroded that you can never get it off again. That has been the case in most apartments I’ve been in (that the screws are so corroded that they can’t be removed).
Just unscrew the old one, find a small strainer and useone of these. Easy peasy and cheap.