I don't know this phrasal verb but you can say to get better, to recover, to get over .
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YankeeThank you Doll and Yankee.
I might say it this way:
Tom was coming along quite nicely for a while, but then he/his health started deteriorating again.
YankeeHi Gary
You can use 'come along' to mean 'successfully advance or proceed'. So, in your context, 'come along' could also mean 'improve'. But your sentence was awkward. In the context you suggested, it would be more typical to use 'come along' in a continuous tense followed by 'quite + adverb'. That's why I suggested an alternative.