Variation of metric

I am looking at the derivation of the Einstein field equations as the Euler-Lagrange equations of the Hilbert functional.
To do this one starts with a variation
$$ g(t) = g+th $$
of the metric, where $h$ is a symmetric 2-covariant tensor.
For small $t$, $g(t)$ will be invertible (if we interpret it as a matrix), so it makes sense to consider the components $g(t)^{ij}$ of the inverse. We have
$$ 0 = frac{d}{dt}Big|_0 (g(t)_{ij} , g(t)^{jk}) = h_{ij} g^{jk} + g_{ij}h^{jk} quadRightarrow quad h^{lk} = – g^{il} g^{jk} h_{ij}. $$
My question: What exactly do the coefficients $h^{lk}$ represent?

I always thought that if one has a, say, 1-covariant tensor $A = A_i dx^i$, then $A^i$ denote the components of the 1-contravariant tensor $A^#$ (cf. musical isomorphisms), given by $A^i = g^{ij}A_j$. But in the formula for $h^{lk}$ above we also have a minus sign in front, so as far as I can see the coefficients $h^{lk}$ are not obtained by raising the indices of $h_{lk}$. That being said, the metric is not really fixed in this case so the musical isomorphisms aren’t either, which may be the source of confusion for me.