Based on this article:
When should one build the Sukkah?
The pious are careful to build to begin building the Sukkah on Motzei Yom Kippur and finish it the next day in order to go from one
mitzvah (Yom Kippur) to another (Sukkot).
If there is a shabbat between Yom Kippur and Sukkot some poskim say that there is an obligation to build it before that shabbat. Avnei
Nezer 459 writes that this obligation isn't just because of zrizin
makdimin limitzvot and therefore it is an obligation. However, the
Minchat Elazar 4:55 writes that this is not an obligation but just the
preferred option, and this is the conclusion of Yalkut Yosef page 118
and Chazon Ovadia Sukkot page 99.
Preferably, one should build it for himself, but if this isn't possible he should appoint an agent to do it for him and strive to at
least take part in the building in some form even if this means
missing out on more learning time.
If a Sukkah was set up with the S'chach from before 30 days before sukkot and it wasn't made for the purpose of the mitzvah of Sukkot one
must change or improve one thing (such as putting down S'chach) of an
area a Tefach by a Tefach or the whole width even if it's of minimal
length (or vice versa). However, if it was made within 30 days of
Sukkot or was made for the express purpose of Sukkot it's fit even
without any change. However, if the Sukkah was originally built for
Sukkot and stands unchanged year to year it requires some change the
each year. This is only if the S'chach was put in advance but if just
the walls were putup in advance and not the S'chach there's no issue.
From these rules, I infer that it is preferable to start after Yom Kippur, but there doesn't seem to be a problem starting before. One year, my shul put up the succah before Tish'a B'Av. Yes, I thought it was bizarre, but they explained that the shul's maintenance supervisor, who constructed the succah each year, was going a long vacation, and he wasn't planning on returning until just before Succot, not leaving enough time to do it. (Separate question, perhaps, re the maintenance guy wasn't Jewish...)
My understanding is that the succah isn't complete (i.e., it's not an official Succah) without the schach. So, based on rule 4, above, I believe that if you put the scach on after Yom Kippur, you're fulfilling the technical terms of "building a succah", no??