Both Uses

on the other hand
in
The Killer Angels
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But then, on the other hand, we sure do stomp them consistently, now don't we, Pete?†
Chpt 1.4 *
Chamberlain said no thanks, wondering how to conquer pride and if a general would part with some chicken, and then felt ashamed, because his boys had none and would be guilty to eat something up here, but on the other hand, don't get something soon, and keep losing blood, might pass out, in all this damned heat, like you did the other time, and be no good to anybody.†
Chpt 3.3

Definitions:

(1) (on the other hand)
from another point of view; or in a way that is different (a phrase used to introduce a different perspective or idea)
(2) (meaning too rare to warrant focus)