What I have in mind is something like the following: the natural numbers with addition form a monoid. You can imagine constructing the integers by taking the naturals, adding a set constructed by "reflecting" all of the non-zero natural numbers about $0$, and extending addition. What I mean by "reflecting" is that for each $i\in N$ the reflected element $-i$ is such that $i+j=k\iff -i+-j=-k$, and that $i+-i=0$.
My intuition is that this should be generally possible for monoids that don't have any inverses.
This is possible for commutative, cancellative monoids (cancellative means $a+b = a+c$ implies $b=c$, which is automatic in a group but not a monoid). It's essentially the same as constructing $\mathbb{Z}$ from $\mathbb{N}$, as you said. But otherwise it doesn't quite work.