One way to generalize your theorem is to bound the spherical curve with a circle of given (radial) diameter $D$.
So the generalization of your theorem:
"If a spherical curve fits within a hemisphere without touching a great circle, then two copies of the curve can be fit on the sphere without intersecting."
Will be:

"If a spherical curve fits within a circle (or "cap") of radial
diameter $D$, then $N$ copies of the curve can be fit on the sphere
without intersecting. Where $N$ is defined as below"

If we want, this condition can also be formulated in terms of $\theta(t)$ and $\phi(t)$, since a circle on a sphere with radial diameter $D$ can be defined by an intersection with a cone centered at the origin.
Namely, if there exists a unit vector $\overrightarrow v$ such that the dot product between $\overrightarrow v$ and any vector
$(x(\phi,\theta), y(\phi,\theta), z(\phi,\theta))$ is larger than $\cos(D/2)$, then the curve is within the circle (see 2d illustration figure below).

The following simple argument gives a bound on $N$.
Let $D$ be the radial diameter of the circle, if
$D \leq 2\pi/n$ then the number of copies $N$ satisfies $N \geq 2n-2$ for even $n$, and $N \geq 2n-3$ for odd $n$.

To see this we pack the circles with their centers on the equator (a great circle) $2\pi/n$ apart, which gives us $n$ copies.
We then place the other circles in a similar manner on a great circle that is perpendicular to the equator.
For even $n$ it is easy to see that $n-2$ circles can still be placed along the perpendicular great circle, since on both sides of the strip of (radial) width $2\pi/n$ there remains two arcs spanning $\pi - 2\pi/n$ each, which can thus hold $2(n/2-1) = 2n-2$ copies of the circle.

For an odd $n$ the configuration is a bit more complicated but with a similar argument we can see that there remains room for $2n-3$ copies (of the odd $n$ circles that can be placed on a great circle in general, one is blocked from one side and two are blocked from the other, see illustration below).

The following figure gives a 2d illustration of these configurations for $n=6$ and $n=7$ (the two green lines bound the strip that is blocked by the circles packed along the equator).

We can see that this definition of $N$ already generalizes your theorem, which is a private case for $D=\pi$.
For $D=\pi$, $n=2$ for which $N=2n-2=2$ as the theorem states.

However, we can do better than that. It turns out that we can view this as an inverse of a known packing problem "Tamme's Problem".
Whereas Tamme's problem is given a number $N$ and asks what is the maximal diameter $D$ for which $N$ circles can be packed on the sphere, our problem is given a diameter $D$ and asks what is the maximal number $N$ of circles that can be packed on the sphere.

While there are optimal solution to Tamme's problem for $N \leq 12, N=24$, there is no known optimal
solution for general $N$ (see this paper and its references for example).
Still, there are solutions for $N$ values up to 150, which are conjectured to be optimal.

**So to conclude**, your theorem can be generalized to any circle bounding the curve, and the problem can then be mapped to the inverse of Tamme's problem.
The number of copies $N$ depends on the solutions to Tamme's problem (and is at least $2n-3$ for $D \leq 2\pi/n$).
Given a circle of diameter $D$ that bounds the curve, one can look up in a table of Tamme's solutions (such as *Table 1* from here) for the maximal $N$ for which the known solution to Tamme's problem is still smaller than $D$.