The quasi-transverse-momentum dependent (qTMD) distributions are equal-time correlators that can be computed within the lattice QCD approach. In the regime of large hadron's momentum, qTMD distributions are expressed in terms of standard TMD distributions via the factorization theorem. We derive the corresponding factorization theorem at the next-leading power (NLP), and, for the first time, we present the factorized expressions for a large class of qTMD distributions of sub-leading power. The NLP expression contains TMD distributions of twist-two, twist-three, and a new lattice-specific nonperturbative function. We point out that some of the qTMD distributions considered in this work can be employed to extract the Collins-Soper kernel using the standard techniques of different-momenta-ratio. We provide NLO expressions for all the elements of the factorization theorem. Also, for the first time, we explicitly demonstrate the restoration of boost invariance in NLP TMD factorization.