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."
"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 $D$.