だから僕は、たまに彼を昔の僕と重ねていた。
重ねて:
1. 同じ行為・事態をもう一度繰り返すさま。再び。文章語で、多く手紙や公式の発言に用いる。 「 －要望する」
2. この次。今後。 「 －は泥鰌（どじよう）にてもあれ、鮠（はえ）にても候へ必ず持つて伺候致さうずる／狂言・鱸庖丁」
(Source)
The context is the narrator finds his classmate quite similar to his former self. So point 1 fits the above sentence.
Could I translate it as follows:
That's why I sometimes saw my former self again in him.
Thank you for the guidance.
だから僕は、たまに彼を昔の僕と重ねていた。
重ねて in 重ねていた is not an adverb but the te-form of the verb 重ねる.
Here (～を‥と/に)重ねる means "to identify ~~ with‥" "to see ~~ in‥" (≂重ね合わせる), hence the translation:
That's why I sometimes saw my former self in him.
This 重ねて is a simple te-form of the verb 重ねる, followed by いた that makes the past-progressive form (expresses a past habitual action in this case). Or you can say this is the past-teiru-form of 重ねる. Here, 重ねる means "to identify with someone" or "to project someone into ～".
だから僕は、たまに彼を昔の僕と重ねていた。
That's why I sometimes projected him into my past self.
As you have cited, 重ねて also has a lexicalized usage ("once again"), but you can forget it for now.
