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Yes, it is called the cofinality of the order. Just apply Zorn's lemma to the class of all well-ordered suborders of $P$, ordered by end-extension. If you've got a maximal such order, then it must be cofinal, since otherwise you could end-extend it. Note that Zorn's lemma applies, since the union of chain of end-extensions of well-orders is still a well-order.