If there's a word, it's probably very obscure. I'm finding no suitable synonym for "second" that conveys that formal connotation.
It may be tempting to some to think there's a word that looks like "penaugural" or "secundaugural" or some such. But "inaugural" comes from the Latin root inaugurare, meaning to install. It's probably not all of a piece - inaugurare also means to practice augury, or to take omens from the flight of birds, but this doesn't lend itself to taking a different affix to denote doing it a second time.
People have tried. "Biaugural" does appear here and there, and appears to be intended to mean what you want - the second occurrence of a formal event, or instance of a formal entity. But it's not in OED, and it's not in Ngrams. In Latin, it would roughly translate as "two omens".
(Of course, if you're being cheeky and informal, you could coin your own term, like "secundaugural" above, or even "deuteraugural" if you really want the word nerds to grind their teeth. ;-) )
"Secondary", unfortunately, is more likely to mean "second place in importance" than "second place in time of occurrence", so I wouldn't use it.
Since your event is annual, you could almost use an anniversary word for it, which would make the second class a biennial. However, this virtually always means "occurring every two years", which is also not what you want. Sufficiently unambiguous and formal, however, is the following:
second annual graduating class