Note that if $\frak m\leq^* n$, then $2^\mathfrak m\leq 2^\mathfrak n$, since given a surjection from $N$ onto $M$, the preimages define an injection between the power sets. So, if $\frak m\leq^* n\leq^* m$, then $2^\mathfrak m=2^\mathfrak n$.

In particular, if there is a Dedekind-finite cardinal, then there is one for which $1+\frak m\leq^* m$. Therefore, if there are any Dedekind-finite cardinals, then the first implication fails to reverse.

Interestingly, $\aleph_0\leq^*\frak m$ is a weaker condition than $1+\frak m\leq^*m$ (so the former does not imply the latter).

Note that if $\frak m\leq^* n$, then $2^\mathfrak m\leq 2^\mathfrak n$, since given a surjection from $N$ onto $M$, the preimages define an injection between the power sets. So, if $\frak m\leq^* n\leq^* m$, then $2^\mathfrak m=2^\mathfrak n$.

In particular, if there is a Dedekind-finite infinite cardinal, then there is one for which $1+\frak m\leq^* m$. Therefore, if there are any Dedekind-finite cardinals, then the first implication fails to reverse.

Interestingly, $\aleph_0\leq^*\frak m$ is a weaker condition than $1+\frak m\leq^*m$ (so the former does not imply the latter).