There's an inversion that happens in these tags at the end of a sentence: "I do something and so does he." The first clause is subject + verb and the second clause is verb + subject. And so it feels like (this means I haven't looked this up so I may be wrong) this inversion is the reason the adverb is in a different position. In your second sentence, once you have an object the word order is regular again.
Student or Learner