Upcoming mass production of energy efficient spin-transfer torque magnetoresistive random access memory will revolutionize microelectronics by introducing non-volatility not only in memory but also in logic. The pressing issue is to boost the sensing margin by improving the tunneling magnetoresistance ratio. We demonstrate that spin-dependent trap-assisted tunneling in magnetic tunnel junctions can increase the tunneling magnetoresistance ratio. The impact of spin decoherence and relaxation on the current and shot noise at trap-assisted hopping is investigated in both normal contact-trap-ferromagnet and ferromagnet-trap-ferromagnet systems. In addition, our approach resolves a controversy between the two theoretical approaches to spin-dependent trap-assisted tunneling available in literature.