How to "switch" a vector field from $TM$ to $T^*M$

Question

Let $(M,g)$ be a Riemannian manifold. Then the metric provides musical isomorphism $\flat:TM\rightarrow T^*M$. Now, consider a vector field on $TM$, $X:TM\rightarrow TTM$; is there a way to induce an associated vector field $\tilde{X}:T^*M\rightarrow TT^*M$ through the musical isomorphism?