This is a partial answer. We'll reduce the question to one of representation theory (of Lie groups), and give an answer in two "extreme" cases.
Suppose $\nabla$ is the Levi-Civita connection of some Riemannian metric $g_1$. Since $\nabla$ is torsion-free, we know that $\nabla$ will be the Levi-Civita connection of a metric $g_2$ if and only if $\nabla g_2 = 0$. This means that we have to understand which (positive-definite) symmetric $2$-tensor fields are $\nabla$-parallel. Understanding which tensor fields are $\nabla$-parallel can be accomplished via:
The Holonomy Principle: Let $\nabla$ be a connection on a connected smooth manifold $M$. Let $\text{Hol}_x \leq \text{GL}(T_xM)$ denote the holonomy group (really, holonomy representation) of $\nabla$ at $x \in M$.
(a) If $T \in \Gamma(TM^{\otimes r} \otimes T^*M^{\otimes s})$ is a parallel tensor field on $M$, then $T|_x$ is fixed by the $\text{Hol}_x$-action on $T_xM^{\otimes r} \otimes T_x^*M^{\otimes s}$.
(b) Conversely: If $T_0$ is a tensor at $x$ fixed by the $\text{Hol}_x$-action on $T_xM^{\otimes r} \otimes T_x^*M^{\otimes s}$, then there exists a unique parallel tensor field $T$ on $M$ with $T|_x = T_0$.
Since our connection $\nabla$ is the Levi-Civita connection of some (let's say Riemannian) metric $g_1$, we have $\text{Hol}_x \leq \text{SO}(T_xM, g_1) \cong \text{SO}(n)$. The question is now: What is the space of (positive-definite) symmetric $2$-tensors at $x$ which are fixed by the $\text{Hol}_x$-action on $\text{Sym}^2(T^*_xM) \subset T_x^*M^{\otimes 2}$. This is a question of representation theory.
Example: (The trivial example) Suppose $\nabla$ is the Levi-Civita connection of a flat metric $g_1$, and suppose $M$ is connected and simply-connected. Then $\text{Hol}_x = 0$ is the identity group, so every element of $\text{Sym}^2(T_x^*M)$ is fixed.
Concretely: If $g_0$ is any (positive-definite) symmetric $2$-tensor at $x$, then (by the Holonomy Principle) we can extend $g_0$ uniquely to a $\nabla$-parallel tensor field $g$ on all of $M$. The upshot is that, for the sort of connections $\nabla$ in this example, we essentially have an $\binom{n+1}{2}$-dimensional space of compatible metrics.
Note that the dimension $\binom{n+1}{2}$ is the largest possible. For the Levi-Civita connection of a "generic" Riemannian metric $g_1$, the holonomy group will be all of $\text{SO}(n)$, and the space of compatible metrics will be $1$-dimensional.
I do not know about the intermediate cases -- i.e., when $\nabla$ is not generic and not flat. (My representation theory needs some work!)
