Two piece jambs aren't all that uncommon, often a builder or installer gets a standard depth jamb when deeper is actually needed, so they'll add another extension to that. "Screwing" something over the top would be unconventional, but adding on to the front face inward isn't entirely uncommon. Best to step it down slightly so it looks like its by design as opposed to trying to hide a seam that can't be hidden. What are you doing for casing? There are pre-built casing/jamb box kits that can be sourced as well, that may be a nice option here.

Appearance would be the only caveat.

I would nail them on , faster and easier then screws and less like chance of having to fill in holes from the screw heads if they are showing.

Hi. I am also planning to soundproof two walls of a bedroom facing a busy highway. However, before playing with the walls insulation, I would like to check how much noise reduction I can get by adding acrylic inserts on the windows. The problem is that I have no extra space behind the windows to create air dead volume (windows are almost flushing the inner walls). I found this post while looking for how to extend the jambs :). I was wondering how much noise reduction you got at the end of our project. Thank you.