What is a "microglot"?

@Greybeard's analysis of the etymologies and evolutions of the terms that make up this expression is, as always, thorough and illuminating.

I'd like to add information about where they are actually in current usage.

Microglot is a term that is apparently used frequently by what I'd call "language hobbyists", who seriously pursue languages and linguistics either in the context of or outside their professional careers. There are a number of online communities, including the Polyglot Conference, where people who love learning about languages congregate and share research they've pursued. In these communities, microglot seems to be commonly used to describe people who study languages that are endangered or whose use is localized to small communities or regions. It's still unclear to me whether J.L. Austin actually coined the term in A Plea for Excuses, or whether it had been used previously.

Draw the coverts is an expression taken from fox-hunting terminology. (See also Fox Hunting Life.)

A covert is a thicket or other naturally formed configuration where a fox might take shelter. It's pronounced "cover". When hounds search through a covert to find a fox and get it to emerge, it's called "drawing the covert" or "drawing covert".

With this information, I come to roughly the same conclusion as Greybeard regarding the meaning of Austin's phrase in the OP's question about drawing the coverts of the microglot. It's an elegant way of characterizing the activity of searching for and ultimately extracting the most elusive and intricate features of language, an activity that perhaps leads to the same joy and fascination experienced by people who study obscure languages not yet explored.