Let us assume (probably wrongly) that P=NP, meaning that we know a way to output a hampath if it exists. 

A graph has $n$ vertexes.

Can the algorithm that solves the problem be modified in polynomial time to an algorithm that solves the problem sequentially?

By sequentially, I mean that the algorithm solves the problem by this way: an algorithm first finds whether a path with two edges that do not return to any of vertexes already reached exists, and then uses information gathered in finding aforementioned case to find whether a path with three edges that do not return to any of vertexes already reached exists and so on until $n$ vertexes.