This is a follow up to my previous question on Silver indiscernibles.
Background: Suppose that $0^\#$ exists, $\alpha<\lambda$ are limit ordinals, $i_\alpha$ is the $\alpha$th Silver indiscernible, and $\lambda$ is contained in $L_{i_\alpha}$ (which is just the model generated by the first $\alpha$ indiscernibles). In $V$, we can define an injection from $\lambda$ to $\omega\times\alpha^{< \omega}$ by simply specifying which term and indiscernibles generate each member of $\lambda$.
Question: Suppose that $\lambda$ is not a Silver indiscernible, is there also a constructible injection from $\lambda$ to $\omega\times\alpha^{< \omega}$?
Edit: By Francois' answer to my previous question, I can now see that the injection described in the background can't be in L (as the union of members in the range of the function will give an infinite constructible set of indiscernibles), so I just want to emphasize that any constructible injection to $\omega\times\alpha^{< \omega}$ (or just to $\alpha$) would be fine as an answer.
Edit #2: Due to the answer written by Apostolos, I've added the assumption that $\lambda$ is not a Silver indiscernible.