  This paper continues the study of the Ramsey-like large cardinals.
Ramsey-like cardinals are defined by generalizing the characterization of
Ramsey cardinals via the existence of elementary embeddings. Ultrafilters
derived from such embeddings are fully iterable and so it is natural to ask
about large cardinal notions asserting the existence of ultrafilters allowing
only $\alpha$-many iterations for some countable ordinal $\alpha$. Here we
study such $\alpha$-iterable cardinals. We show that the $\alpha$-iterable
cardinals form a strict hierarchy for $\alpha\leq\omega_1$, that they are
downward absolute to $L$ for $\alpha<\omega_1^L$, and that the consistency
strength of Schindler's remarkable cardinals is strictly between 1-iterable and
2-iterable cardinals.
