In favorable circumstances we expect low order equations in $I_\gamma$ to span $I(C)$.
For example, smooth rational curves of degree up to 4 in K3s.
We need "isolated" classes, so elliptic curves are also hard. An elliptic curve $C$ has $[C]^2 = 0$ and moves in a pencil, so its periods do not determine a single curve.
No hope to recover rational curves of degree higher than 4. For $d \geq 5$ one needs $a \geq 3$ before $I(C)_a$ is non-zero, and there the Jacobian ideal contributes superfluous equations, so $I(C)_a \subsetneq I_{\gamma,a}$.
For a smooth rational quartic curve $C \subset X$ we have that the equation of the quadric surface containing $C$ generates $I_{[C],2}$, i.e., $I(C)_2 = I_{[C],2}$.