The trivial necessary and sufficient condition is that the initially given external measure
is the external measure of some sigma-measure. In other terms, a closed object for the
Galois correspondence between positive sigma-measures (on sigma-algebras) and positive
external measures (defined on all subsets). Is this trivial condition completely useless?
Well, you learn that "regular" external measures are the essentially same as "comp0lete"
measures. You can also obtain that when you start from a finitely additive measure on a
semiring, or a lattice, of sets, then all the initially given measurable sets (and even
the Peano - Jordan measurable ones) are still measurable for the sigma-measure that
you obtain with the two-step Caratheodory process. Some books on measure theory
also consider the case of a initially given finitely subadditive measure
(on a ring of sets).
There is another process, again with a Galois correspondence, to "bi-complete" a measure:
not only you want that a subset of a zero-measure set again has measure zero, but also
that if a set has infinite measure then it has a subset with finite nonzero measure (or,
equivalently, the measure is the sup of the measure of the integrable subsets).
I think that standard measure theory text should have all this (Fremlin should be freely
available online). A more sophisticated question is the distinction between the above
measures (complete and locally finite) and the measures that are direct sum of finite
measures. A sigma-finite and complete measure is a direct sum of finite complete measures
and a direct sum of finite measures is complete and locally finite, but neither of
the implications can be reversed (the counterexamples for the first are very easy,
for the second they are not). In standard books, you can check also the relation of
the above conditions with the following possible properties of a complete (positive
sigma-additive) measure: (a) the Boolean algebra of measurable sets modulo measure zero
sets is complete; (a') an analogue for the vector lattice of measurable functions
modulo almost everywhere zero functions; (b) the natural duality between integrable
and bounded measurable functions gives the dual of the Banach space of integrable
functions (modulo "almost everywhere". The dual of bounded measurable functions is never
reduced to the integrable ones [i.e. absolutely continuous sigma-measures],
except for trivial [finite dimensional] cases; the dual of the other usual spaces
is the "expected one" for all complete measures); (c) there is a "lifting" for
the bounded measurable functions modulo "almost everywhere" to the bounded measurable
functions. [Hint: (c) is subtly different from the preceding properties, check
in the standard books]
I added this last part about "locally finite" measures becouse it is related to the
other question of yours, but note that it is not exactly the same as the Bourbaki
distinction (some measure theory books write about the distinction between the
tau extension and the sigma-extension for a Daniell integral). I hope that with all
these keywords you can check in your measure theory books.