Let $T$ be a Suslin tree and $f:T\to Y$ be continuous. ($T$ is endowed with the order topology.) Assume that the image of $T$ is contained in a Lindelöf subset of $Y$. Then, force with $T$. Which properties of $f$ and/or $Y$ ensure that the image of $T$ is still contained in a Lindelöf subset of $Y$ in the forcing extension ?

I know that spaces which remain Lindelöf after a countably closed forcing are called indestructibly Lindelöf and have been recently well studied. The preservation of Lindelöfness after adding Cohen or random reals has also been investigated, with some interesting results. It might be that some of these results hold as well when one forces with a Suslin tree, but I must admit that my knowledge of forcing is not very "hands on" and I am not able to see quickly if this is the case.

My motivation is to try to extend an old result of Steprans which says that a continuous real valued map on a Suslin tree has a countable image.