By the 1956 season, Astroth was pretty much done as a major league player. His Baseball Reference major league page shows a few games in Kansas City, but no transactions to explain his appearances in San Diego later that season, or in Buffalo (back in the A's system) in 1957-8.
We find one of the answers in the San Rafael, CA Daily Independent Journal of May 17, 1956...
"Kansas City Athletics—In addition to the Santiago and Valo moves, optioned pitchers Arnie Portocarrero to Birmingham of the Southern Assn., Art Ceccarelli to Columbus of the International League and Bill Harrington to San Diego of the Pacific Const league, and sold catcher Joe Astroth to San Diego."
So the move was not an option, it was a straight out sale to San Diego, an independent team at the time.
The next season, former-Illini Astroth ends up in Buffalo. From the Reading Eagle of Oct. 13, 1956...
“San Diego of the Pacific Coast League sold three players today….Padre General Manager Ralph Kiner said he sold catcher Joe Astroth to Buffalo of the International League…”
A straight-up sale from a then-independent team back to the A's organization, AAA Buffalo. The article mentions that Cleveland was about to take over the Padres for their AAA team. Good to be able to wrap up the transactions for the former Illini athlete.