In this paper, we propose a framework for studying the properties of the Lefschetz thimbles decomposition for lattice fermion models approaching the thermodynamic limit. The proposed set of algorithms includes the Schur complement solver and the exact computation of the derivatives of the fermion determinant. It allows us to solve the gradient flow equations taking into account the fermion determinant exactly, with high performance. Being able to do so, we can find both real and complex saddle points and describe the structure of the Lefschetz thimbles decomposition for large enough lattices which allows us to extrapolate our results to the thermodynamic limit. We describe the algorithms for a general lattice fermion model, with emphasis on two widely used types of lattice discretizations for relativistic fermions (staggered and Wilson fermions), as well as on interacting tight-binding models for condensed matter systems. As an example, we apply these algorithms to the Hubbard model on a hexagonal lattice. Several technical improvements allow us to deal with lattice volumes as large as 12×12 with Nτ=256 steps in Euclidean time, in order to capture the properties of the thimbles decomposition as the thermodynamic, low-temperature, and continuum limits are approached. Different versions of the Hubbard-Stratonovich (HS) transformation were studied, and we show that the complexity of the thimbles decomposition is very dependent on its specific form. In particular, we provide evidence for the existence of an optimal regime for the hexagonal lattice Hubbard model, with a reduced number of thimbles becoming important in the overall sum. In order to check these findings, we have performed quantum Monte Carlo (QMC) simulations using the gradient flow to deform the integration contour into the complex plane. These calculations were made on small volumes (Ns=8 sites in space), albeit still at low temperatures and with the chemical potential tuned to the van Hove singularity, thus entering into a regime where standard QMC techniques exhibit an exponential decay of the average sign. The results are compared versus exact diagonalization, and we demonstrate the importance of choosing an optimal form for the HS transformation for the Hubbard model to avoid issues associated with ergodicity. We compare the residual sign problem with the state-of-the-art BSS (Blankenbecler, Scalapino, and Sugar)-QMC and show that the average sign can be kept substantially higher using the Lefschetz thimbles approach.