Bostick cleared waivers and was outrighted the Triple-A Syracuse on Saturday, Deesha Thosar of the New York Daily News reports.
The Cardinals signed Bostick to a minor-league contract Tuesday.
Bostick was released by the Cardinals in March, Baseball America reports.
Bostick signed with the Cardinals as a non-roster invitee Friday.
Bostick was released by the Astros on Saturday, Chandler Rome of the Houston Chronicle reports.
Bostick was placed on the 7-day disabled list Saturday with a hand injury, Brian McTaggart of MLB.com reports.