As one class of binary continuous aggregation functions which play an important role in practical applications, overlap functions defined on the unit closed interval have been developed rapidly in the past decade. At the same time, Paiva et al. recently extended the concept of overlap functions on unit closed interval to the lattice-valued status and called them quasi-overlap functions on bounded lattices. In this paper, we mainly study the extension methods of quasi-overlap functions and their three generalized forms on bounded partially ordered sets. More concretely, first, we show some new extension methods of quasi-overlap functions, 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions on any bounded partially ordered set P by the so-called ⋄-operators and 0,1-homomorphisms, 1-⋄-operators and 1-homomorphisms, 0-⋄-operators and 0-homomorphisms, and 0,1-⋄-operators and ord-homomorphisms, respectively, which are different from the extension methods obtained by Qiao lately. And then, as an application of the new extension methods, some concrete quasi-overlap functions, 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions on some certain bounded partially ordered set P are constructed. Finally, we prove that these extensions maintain idempotent and Archimedean property of the known quasi-overlap functions, 0P-quasi-overlap functions, 1P-quasi-overlap functions and 0P,1P-quasi-overlap functions on any bounded partially ordered set P.