Generally speaking (modulo some overlap), there are two types of finiteness properties for a finitely generated group $G$, homotopical properties (e.g. type $F_n$, type $F$, $cd_\mathbb{Z}<\infty$) and group theoretic ones (e.g. linearity, residual finiteness, Hopficity, being virtually soluble).
We can group some of these into 'families', such as $FP$, $FP_\infty$ and $FP_n$. Or linearity, residual finiteness and Hopficity.
It is well known that: Linear $\implies$ Residually Finite $\implies$ Hopfian.
My question is whether there is a stronger property which implies linearity or a weaker property that is implied by Hopficity, but is still stronger than just being finitely generated. Preferably the property appears in the literature and there are examples to show the property is strictly stronger or weaker (by examples, I mean something like Drutu and Sapir's group with is not linear, but is residually finite).
For a stronger property, 'virtually special' is one such property which appears in literature, and is strictly stronger than linearity. By virtually special I mean that the group $G$ has a finite-index subgroup $H$ which is the fundamental group of a special cube complex. This implies that $H$ is a subgroup of a Right-Angled Artin Group, and $G$ is hence a subgroup of $SL_n(\mathbb{Z})$ for some $n$. This property is strictly stronger than just being linear over $\mathbb{Z}$, see for example https://arxiv.org/abs/1402.6974, which implies that $SL_3(\mathbb{Z})$ is not virtually special. Whether it's fair to call this a 'finiteness' property as such is debatable, but it fits the criteria of property you were asking for, especially in that it isn't just an arbitrary extension of previously mentioned properties.
For a weaker property, I may suggest 'sofic'. It is not known whether all finitely generated groups are sofic or not, so I can't provide an example showing that it strictly fits your criteria, however it is fair to call this a 'finiteness' property and this hasn't been mentioned yet in the comments.