  Numerical simulations of quantum chromodynamics (QCD) on a lattice require
the frequent solution of linear systems of equations with large, sparse and
typically ill-conditioned matrices. Algebraic multigrid methods are meanwhile
the standard for these difficult solves. Although the linear systems at the
coarsest level of the multigrid hierarchy are much smaller than the ones at the
finest level, they can be severely ill-conditioned, thus affecting the
scalability of the whole solver. In this paper, we investigate different novel
ways to enhance the coarsest-level solver and demonstrate their potential using
DD-$\alpha$AMG, one of the publicly available algebraic multigrid solvers for
lattice QCD. We do this for two lattice discretizations, namely clover-improved
Wilson and twisted mass. For both the combination of two of the investigated
enhancements, deflation and polynomial preconditioning, yield significant
improvements in the regime of small mass parameters. In the clover-improved
Wilson case we observe a significantly improved insensitivity of the solver to
conditioning, and for twisted mass we are able to get rid of a somewhat
artificial increase of the twisted mass parameter on the coarsest level used so
far to make the coarsest level solves converge more rapidly.
