If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the "standard" natural numbers, apparently an application of downward Lowenheim-Skolem theorem should yield a countable nonstandard model $\mathcal{N}'$ of PA (such as the one described by Skolem in 1933/34). Is this a correct application of the theorem? Not being an expert in the field I am concerned there might be some details I overlooked.

Second, if one wants to get more of the structure available in Skolem's model, such as transfer for all definable functions, how would one do this exactly using downward L-S?