There is one due to Satô, "Almost analytic vector fields in almost complex manifolds". He constructs a $J'$ and $J''$ on $T^\ast M$ using a $J$ on $M$, such that $J''$ is always an almost complex structure but $J'$ is only an almost complex structure when $J$ is integrable (in which case $J'$ is also integrable), and $J''$ is a modified version of $J'$ using the nontrivial Nijenhuis tensor of $J$. This is elaborated on in Yano--Patterson's paper "Vertical and complete lifts from a manifold to its cotangent bundle" (which might be easier to digest).
There is also a way using connections on $M$ (which generalizes Satô's work) due to Yano--Patterson, "Horizontal lifts from a manifold to its cotangent bundle". Then there is a unifying generalization of both works, see Bertrand's paper "Almost complex structures on the cotangent bundle" (https://arxiv.org/abs/math/0507068); here Theorem 3.1 asserts holomorphicity of the projection $\pi$.