Law of total probability in its mainstream formulation assumes a partition ($B_i$'s) of the universe $\Omega$ to compute the probability of any event $A$ as
$$\mathbb{P} (A)=\sum_i \mathbb{P}(A \cap B_i).$$
But, can't we weaken the hypothesis by only asking $A \subseteq \bigcup_i B_i (\subseteq \Omega)$ while still, of course, asking the $B_i$'s to be disjoint (just relax the hypothetis that $\bigcup_i B_i = \Omega$, by only keeping an inclusion relation) ?
By doing so, if the $B_i$'s are indeed a partition of $\Omega$, well that will work, but it allows to handle more general situations where some events of $\Omega$ might be outside the $B_i$'s AND not intersect $A$ neither (in which case, we don't need to care about them in the computation of $\mathbb{P} (A)$) .
What do you think about it ?
This follows at once from the usual rule.
Indeed, let $\overline B$ denote the complement of $\bigcup B_i$ and adjoin $\overline B$ to your collection. Now you have a true partition and the assumption that $A\subseteq \bigcup B_i\implies A\cap \overline B=\emptyset$.
Now you can invoke the standard rule to reach the desired conclusion.