First, introductory, inaugural...is there another term referring to the item coming afterward?
Context: I attended the second graduating class of a school that started a year before I joined. I can't say I was in the inaugural class of that school, because I was in the second class. If I was in the last class, I could say "ultimate" (or "last") class...if I was in the second-to-last class, I could say "penultimate" class.
Is there another word for "second" in this case?
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
Greek letters are sometimes used for this purpose; you would call yourself a member of the beta class.
This "numbering" system is used very commonly in sororities and fraternities, which commonly use Greek letters for various purposes, and by extension for other similar organizations.
Thus the first chapter established of an organization with multiple chapters is the alpha chapter, the second is the beta chapter, etc. For example:
Each chapter [of the Phi Beta Kappa honor society] is designated by its state and a Greek letter indicating its position in the order in which that state's chapters were founded. For example, Alpha of Pennsylvania refers to the chapter at Dickinson College, founded in 1887; Beta of Pennsylvania, the chapter at Lehigh University (founded later that same year); Gamma of Pennsylvania, the chapter at Lafayette College (1890); and Delta of Pennsylvania, the chapter at the University of Pennsylvania (1892). ("Phi Beta Kappa", Wikipedia)
And within chapters, the members who joined in the first year would be the alpha class and those who joined in the second year are members of the beta class. For example:
As the second oldest chapter organized in Alpha Kappa Alpha Sorority, Incorporated, Beta Chapter spans across four other Universities, which includes Loyola University Chicago, University of Illinois at Chicago, Columbia College Chicago, and Roosevelt University. ("The Beautiful Beta Chapter", Alpha Kappa Alpha Beta website)
I think in an academic setting this usage would probably be fairly well understood, as for example:
FUV's undergraduate program in engineering and the liberal arts and sciences is scheduled to admit a beta class in 2018. ("Fulbright University Vietnam", Wikipedia)
The sophomore year : (second year)
In the United States, a student in the second year of study at a university or college is called a sophomore. The sophomore year is preceded by the freshman (first) year, and followed by the junior (third) and senior (fourth) years.
