Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected.

I believe it could be proved by directly constructing the ambient framed cobordism (using some tubular neighborhoods and surgery..) but is there not a reference for the claim?

EDIT: $n>0$.