In part I, I considered the problem of discovering when, given an irrational α which has a simple continued fraction representation with convergents pn/qn, there exists α' for which the denominator sequence for convergents is a subsequence of (qn). It was shown that such an α' exists if the continued fraction representation was “nearly periodic” with odd period. The following is a generalization of the results of part I to semi-regular continued fractions, where the problem seems to fit more naturally.