A novel Monte Carlo (MC) simulation scheme based on Theodorou's bijective mapping strategy [D. N. Theodorou, J. Chem. Phys. 124, 034109 (2006)] is introduced. This min-map bias Monte Carlo acts in combination with any other proper, bare MC. It carries over the bare MC move from the original configuration space Omega(0), where trial move acceptance may be low, to a different configuration space, Omega(1), where acceptance is higher. The bare MC move is then performed in Omega(1) and the resulting configuration is finally mapped back to Omega(0). Mappings between Omega(0) and Omega(1) entail weighted selection of trial configurations, the bias of which is subsequently removed in the overall acceptance criterion. The new method is applied, in conjunction with continuum configurational bias as bare MC scheme, to the simulation of explicit hydrogen linear alkanes in the canonical ensemble. Min-map bias MC is found to alleviate the pervasive problem of very low acceptance rates encountered when using an explicit molecular description.