Schrödinger operators with Sturmian potentials have been studied extensively, and a central question, whether every gap predicted by gap labeling actually appears in the spectrum, was recently resolved affirmatively by Band, Beckus, and Loewy. We consider two natural generalizations of Sturmian sequences: codings of rotations and quasi-Sturmian sequences. For both classes, we use the Johnson-Schwartzman gap labeling theorem to identify the set of admissible gap labels; in the quasi-Sturmian setting, this gives the first description of the Schwartzman group for these subshifts. For binary codings of rotations, we go further and show that every admissible label is attained by some Schrödinger operator in the associated family. More precisely, for each label predicted by gap labeling, there exists a sampling function for which the corresponding gap is open.