Almost complex structure parallel with respect to non-Levi-Civita connection

Question

Can an almost Kaehler structure $(M,g,J)$ be shown to be integrable (Kaehler) but using some non-Levi-Civita linear/affine connection? (E.g. finding a $\hat \nabla$ of the form $\hat \nabla = \nabla^{LC} + T$ such that $\hat \nabla J = 0$).