This answer combines my previous comments and adds a little more.

Let $G$ be connected reductive over an algebraically closed field. Let $S \subset T$ be a subtorus of a maximal torus.
I'll denote by $W_S$ the Weyl group $N_G(S)/Z_G(S)$. In particular, $W_T = N_G(T)/T$, as $Z_G(T) = T$.

In general neither of $N_G(S)$ and $N_G(T)$ is contained in the other (it suffices to consider $G = GL_2$). I think the best comparison between $W_S$ and
$W_T$ is the following. Let $W_{S,T} = (N_G(S) \cap N_G(T))/T$. This is clearly a subgroup of $W_T$ and it naturally
maps to $W_S$, as $T \subset Z_G(S)$.

In fact, every Weyl group (in your sense) naturally lives in a maximal one. Note that $Z_G(S)$ is a Levi
subgroup of a parabolic subgroup. (I happen to have Digne-Michel's nice book "Representations of finite groups of Lie type"
at hand, and they prove it in Prop. 1.21.) Moreover, it's clear that $N_G(S)$ normalises even $Z_G(S)$. Thus $W_S$ is
in fact a subgroup of $W_S' := N_G(Z_G(S))/Z_G(S)$. I claim that $W_S'$ is also a Weyl group. To see this, let $M = Z_G(S)$,
a Levi subgroup and let $S' = Z(M)^0$ (the connected component of the centre of $M$). Then $W_{S'} = N_G(M)/M = N_G(S')/Z_G(S')$.
(It's a standard fact that $Z_G(S') = M$, again see [DM], Prop. 1.21; for the second equality note again that $N_G(S') = N_G(Z_G(S'))$.)

Thus the most interesting "Weyl groups" (in your sense) are the ones of the form $N_G(M)/Z_G(M)$ for $M$ a Levi subgroup.
Let $S' = Z(M)^0$ as above. I claim that in this case the natural map $W_{S',T} \to W_{S'}$ is surjective.
Suppose that $n \in N_G(S')$. Then $n M n^{-1} = M$, so $n T n^{-1}$ is a maximal torus of $M$, hence it's of the form
$m T m^{-1}$ for some $m \in M$. It follows that $m^{-1} n \in N_G(T)$, but it's also in $N_G(S')$ (as $m \in Z_G(S')$).
This proves the claim.

Finally, I think it's helpful to look at the simple example of $G = GL_4$ (or $SL_4$), $M$ the Levi with (2,2) blocks containing the
diagonal torus $T$ and $S'$ its centre, that I wrote down in the comments above. In this case $W_{S'} = N_G(M)/M$ is of order two (switching the two blocks),
$W_{S',T}$ is of order 8 surjecting onto it and injecting into $W_T \cong S_4$.

[I didn't pay much attention to positive characteristic while writing this, but I think it should all be fine.]