Almost complex structures on cotangent bundles of almost complex manifolds

Let $$M$$ be a smooth manifold and $$I:TM\to TM$$ an integrable almost complex structure. Then, the cotangent bundle $$T^*M$$ admits a canonical complex structure, which can be built from holomorphic charts on $$M$$.

If $$I$$ is not necessarily integrable, is there always an almost complex structure on $$T^*M$$ such that $$T^*M\to M$$ is holomorphic? Is it canonical? If not, can we construct it using some additional choice on $$M$$?

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 $$\pi$$.