  We study the phase diagram of 5-dimensional $SU(2)$ Yang-Mills theory on the
lattice. We consider two extensions of the fundamental plaquette Wilson action
in the search for the continuous phase transition suggested by the $4+\epsilon$
expansion. The extensions correspond to new terms in the action: i) a unit size
plaquette in the adjoint representation or ii) a two-unit sided square
plaquette in the fundamental representation. We use Monte Carlo to sample the
first and second derivative of the entropy near the confinement phase
transition, with lattices up to $12^{5}$. While we exclude the presence of a
second order phase transition in the parameter space we sampled for model i),
our data is not conclusive in some regions of the parameter space of model ii).
