Toilets, Sinks, Showers, Tubs and Disposals - Tub/shower drain cover

My tub was draining really slowly, so I took the drain cover off, thinking I'd be able to get a snake down there, which, it turns out, is impossible as I'm sure you real plumbers know. Turns out it was clogged with hair, most of which which I pulled out (still a bit slow, think Draincare would help?)
Is there a drain cover available with smaller holes, or even mesh or screen? Seems to me I'd be better off if the hair never got past the drain cover (the holes, at least 3/8 inch in diameter, do strain out some, but not nearly all of the hair).

Not sure about buying a drain cover with smaller holes as they may block the water flow also. What I found, at the Dollar store, was a set of 3 mesh strainers for sinks. 3 different sizes. I use one when I wash my dog in the tub. Stops a lot of hair and its removable when not needed.

If it was just squeaky clean hair snagged up where you can extract it, that's a chore. There normally won't be anywhere else for hair to snag beyond that point. If it was a gooey compound of hair and soap oil caked on then the entire drain line is becoming blocked; use drain cleaning chemicals to clear that, or, use less greasy chemicals when you shower.

Some homes just have this problem of greasy buildup and it must be certain types of hair conditioners or shaving cream or something, who knows?