Combine visitors in VisitorPatern

I implemented the visitor pattern according to cpppaterns.com to traverse a tree. I always need to traverse the tree in the same order, hence I moved the logic which determines which child to visit next, into the base class Visitor. The individual visitors perform specific operations on the tree nodes and then call the visit method in the base Visitor. This works pretty good.

In some cases it is needed to perform actions implemented among multiple visitors (e.g., VisitorRewriteX, VisitorPrintX) on one tree. The naive way is to just execute the visitors sequentially. However, this requires traversing the tree multiple times which is inefficient. Of course, I could also create a new visitor (e.g., VisitorRewritePrintX) which just calls the other two visitors—but I think that's not clean code either. Is there some other way (e.g., pattern) which could help me to somehow allow 'stacking' of the actions implemented in the visitors?