We develop our previous study of the transition to deconfined quark phase in neutron stars, including the interaction in the quark equation of state to the leading order in the perturbative expansion within the confinement density-dependent mass model. Using the Gibbs conditions the hadron-quark mixed phase is constructed matching the latter with the hadron equation of state derived from the microscopic Brueckner-Hartree-Fock approximation. The influence of quark interaction parameters on threshold properties and phase diagram of dense neutron star matter are discussed in detail. We find that the leading-order quark interaction expands the density range of the mixed phase, pushing forward the disappearance of the hadron phase. Moreover, since the equation of state could turn out to be stiffer, a high-mass hybrid star is possible with mixed-phase core with typical parameter sets.