  We study the question up to which power an irreducible integer-valued
polynomial that is not absolutely irreducible can factor uniquely. For example,
for integer-valued polynomials over principal ideal domains with square-free
denominator, already the third power has to factor non-uniquely or the element
is absolutely irreducible. Recently, it has been shown that for any
$N\in\mathbb{N}$, there exists a discrete valuation domain $D$ and a polynomial
$F\in\operatorname{Int}(D)$ such that the minimal $k$ for which $F^k$ factors
non-uniquely is greater than $N$.
  In this paper, we show that, over principal ideal domains with infinitely
many maximal ideals of finite index, the minimal power for which an irreducible
but not absolutely irreducible element has to factor non-uniquely depends on
the $p$-adic valuations of the denominator and cannot be bounded by a constant.
