Franco, who went on the injured list July 10 with a right hamate bone injury that required surgery, hasn't played since leaving his first rehab game Tuesday with the soreness.
The 21-year-old Franco will return to St. Petersburg, Florida, for additional treatment.
No timeframe was announced for when Franco might be able to resume playing.
Franco on Aug. 14 took on-field batting practice for the first time since his surgery. After the session, the switch-hitter said he felt great except when taking right-handed swings.
He also missed 23 games earlier this season with a strained right quadriceps. He has a .260 average with five homers and 23 RBIs over 58 games in 2022.
Franco signed a $182 million, 11-year contract in November that includes a club option for 2033. The deal could be worth up to $223 million with incentives if the club option is exercised.
He hit .288 with 18 doubles, 5 triples, 7 homers and 39 RBIs in 70 games last season and finished third in the American League Rookie of the Year voting. Franco put together a 43-game on-base streak in 2021 to tie Cincinnati's Frank Robinson (1956) for the longest in major league history among players under 21.