  On a closed balanced manifold, we show that if the Chern scalar curvature is
small enough in a certain Sobolev norm then a slightly modified version of the
Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the
Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on
closed almost-Hermitian manifolds, is close enough to a constant function in a
H\"older norm then the Chern-Yamabe problem has a solution for generic values
of the fundamental constant.
