We investigate class of well-poised basic hypergeometric series Jk,i(a;x;q), interpreting these series as generating functions for overpartitions defined by multiplicity conditions. We also show how to interpret the Jk,i(a;1;q) as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases (a,q) →(1/q,q), (1/q,q2), and (0,q), where some of the functions Jk,i(a;x;q) become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.