The excitation gap above the Majorana fermion (MF) modes at the ends of one-dimensional (1D) topological superconducting (TS) semiconductor wires scales with the bulk quasiparticle gap E-qp. This gap, also called minigap, facilitates experimental detection of the pristine TS state and MFs at experimentally accessible temperatures T << E-qp. Here, we show that the linear scaling of minigap with E-qp can fail in quasi-1D wires with multiple confinement bands when the applied Zeeman field is greater than or equal to about half of the confinement-induced band gap. TS states in such wires have an approximate chiral symmetry supporting multiple near-zero energy modes at each end leading to a minigap, which can effectively vanish. We show that the problem of small minigap in such wires can be resolved by forcing the system to break the approximate chirality symmetry externally with a second Zeeman field. Although experimental signatures such as zero-bias peak from the wire ends is suppressed by the second Zeeman field above a critical value, such a field is required in some important parameter regimes of quasi-1D wires to isolate the topological physics of end-state MFs. We also discuss the crucial difference of our minigap calculations from the previously reported minigap results appropriate for idealized spinless p-wave superconductors and explain why the clustering of fermionic subgap states around the zero-energy Majorana end state with increasing chemical potential seen in the latter system does not apply to the experimental TS states in spin-orbit-coupled nanowires.