No issues at all to worry about, in fact many many people do just this. As US passport holders you are visa free so can come and go as you wish. You might asked on your second visa how long you are staying for but that's just for the purposes of the 90 day stay rule and nothing to be concerned about. I've gone in and out 3 times (Moz, Zim, Lesotho) on one trip and not had an issue.

Enjoy Zim. Despite its many challenges it remains a really charming place to travel with some truly wonderful people.