Ancestry.com has a message board intended for census research in Arkansas:

http://boards.ancestry.com/topics.census.us.ar/mb.ashx

Maybe you can find an expert on that board that will do a free lookup in the Federal census of 1920, seeking the MOORE and MATTHEWS families.

The expert might insist on you’re revealing your grandfather’s first name, an item missing from the two queries you posted here on the Arkansas adoption board. The expert might want to know your e-mail address.

Good luck in your research task.

Reg Niles
Author of The Reg Niles Searchbook for Adult Adoptees.
2 volumes (2013). The WorldCat website will list the libraries that purchase the book.