  We study imbalanced fermionic superfluids in an array of one-dimensional
tubes at the incipient dimensional crossover regime, wherein particles can
tunnel between neighboring tubes. In addition to single-particle tunneling
(ST), we consider pair tunneling (PT) that incorporates the interaction effect
during the tunneling process. We find that with an increase of PT strength, a
system of low global polarization evolves from a structure with a central
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state to one with a central BCS-like
fully-paired state. For the case of high global polarization, the central
region exhibits pairing zeros embedded in a fully paired order. In both cases,
PT enhances the pairing gap, suppresses the FFLO order, and leads to spatial
separation of fully paired and fully polarized regions, the same as in higher
dimensions. Thus, we show that PT beyond second-order ST processes is of
relevance to the development of signatures characteristic of the incipience of
the dimensional crossover.
