I think we agree: an implementation that uses a stack to make normal calls fast, needs to move/copy the stack to the heap, when the stack runs out (or when a continuation is captured).

I don't know whether Chez still uses the same strategy, but a major part of Dybvig's thesis (very readable) was dedicated to explain the details of a possible implementation strategy.

http://agl.cs.unm.edu/~williams/cs491/three-imp.pdf