No. That’s just as ambiguous. Other can mean either different or additional.
As Franco-filly says, if you wanted to convey that his financial situation was also good, you would express it differently, so that it clearly had that meaning. However, although “apart from” mainly implies “except for”, if you add “quite” it does heavily imply “in addition to”.
Quite apart from his [very healthy] financial situation, he has a lot else to be grateful for too.
It’s also always interesting to switch clauses around and see what difference that makes. Here, the sentence remains ambiguous, unless you tweak it in such a way as to clarify the meaning:
He has a lot to be grateful for — apart from his financial situation, that is! (money situation bad)
He has a lot to be grateful for, even apart from his financial situation. (money situation good)