A space is submetrizable if its topology contains a coarser metrizable topology.
Each such space is Hausdorff: given two points $x,y$, the sets $B_{|x-y|/2}(x),B_{|x-y|/2}(y)$ are open in the space.
How far can this be strengthened? To $T_3$? To functionally Hausdorff? To $T_{2.5}$?
It cannot be strengthened to $T_3$. Here's an example (which Munkres called $\mathbb{R}_K$). Let $K = \{1/n : n \in \mathbb{N}\}$, and let $\tau$ be the topology on $\mathbb{R}$ generated by the usual open intervals and the sets of the form $(a, b) \setminus K$. Obviously this is finer than the usual metric topology, so it is submetrizable by your definition. But in this topology $K$ is closed and cannot be separated from $0$ by open sets.
It can be strengthened to functionally Hausdorff (and hence also to $T_{2.5}$). In fact, if $\tau_1$ and $\tau_2$ are topologies on $X$ with $\tau_1$ finer than $\tau_2$, and $f : X \to [0, 1]$ is continuous with respect to $\tau_2$, then $f$ is also continuous with respect to $\tau_1$.