Is it possible for a non-towered airport to have keyhole extensions Class E starting at the surface?
There was a control tower at SGH which closed. The FAA issued a rule which removed the Class D airspace as of December 10, 2015.

This leaves the class E airspace chunks. As you noted, the Chart Supplement for SGH has no information about the airspace, and FAA Order 7400.11A (the current version of airspace definitions) states only that, after an initial NOTAM announcing the effective dates and times, (p. E-183) 


  The effective dates and times will thereafter be continuously published in the Airport/Facility Directory.


So, is the airspace between the surface and 700AGL in these two areas class E or G? I'm not a lawyer, but I think you could make a strong argument for the airspace being class G, unless the chart supplement gets an update or a NOTAM is issued.

Finally, to the question: why are these two areas there? Two thoughts come to mind.


The class E extensions existed alongside the class D airspace, and nobody thought to remove them. (This is the most likely, to me.)
There is some other operational reason for the class E airspace. There are instrument approaches to the field, though they terminate well inside the radius of the two airspace segments, so that's not very likely. Also, the class D airspace could be reactivated by NOTAM if a high-traffic event at SGH were to occur; the class E extensions could be part of that. However, both of these options are really stretches - my guess is that we've found airspace that someone just forgot.


2019 update: those extensions are no longer visible on the sectionals:


(skyvector.com)