Wood threw 3 2/3 scoreless innings on Friday, so he would not have been available for the last two games of the Braves' series with the Cincinnati Reds anyway.
Constanza's call up was much needed after Justin Upton, B.J. Upton and Jason Heyward all went down with muscle strains within 27 hours.
The team also told Fox Sports South that they'd like to bring Wood back later as a starting pitcher as opposed to the relief roles he played, so the mandatory 10-day waiting period before they could recall him could help the Braves stretch him back out.
Wood had a 2.45 ERA over 22 innings made in 15 appearances, including one start.
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