$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered.
So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it possible that exist an infinite set $A$ with hartog number $h(A)=\aleph_0$?
Yes. It is consistent to have such sets. These sets are called Dedekind-finite sets.
Do note that $\aleph_0$ is the only minimal infinite cardinal, even if it not the minimum.