No. It only works if the subject of the two clauses is the same. For example: After HE finished the work, HE had a rest = After finishing the work, he had a rest. You can take the shortcut of a reduced clause in this situation, because 'he' is the subject of both parts the sentence. It was the same person who finished the work and had a rest ('had a rest' sounds more natural to me than 'took some rest', but that's a minor point). You can't do this if the subject of the two clauses is different. For example: 'After he finished the work, his girlfriend gave him a big kiss.' If you say 'After finishing the work, his girlfriend gave him a big kiss' this would mean that it was the girlfriend, not the man, who had finished the work.

It's worth pointing out that this construction is optional: your first sentence makes complete sense. "Took some rest" is somewhat unnatural, in this case "he rested" is more normal. We only really use "take a rest" in the imperative mood (telling someone to rest). In general, you can use this kind of construction whenever the subject of the first clause is also the subject of the second clause. We can't use it if the two clauses have different subjects, but English is generally versatile enough to let us rewrite a clause to change the subject if desired. "After eating food, I drank some tea." "After beating Tom at football, I danced" "After being beaten by Tom at football, I cried"