Here’s a summary of what most authoritative Latin grammars say on the genitive singular ending of –io stems (Weiss 2009/2011: 222-223; Leumann 1977: 424-425; Sihler ). For the sake of simplicity and consistency, in my answer I use the periodization of Latin as used in Weiss (which is different from, for instance, Clackson and Horrocks or Meiser). Weiss has a beautiful quote from Wright 2003, "All periodizations are, of course, administrative fantasies."
In Very Old Latin (6th-3rd centuries BCE), the genitive singular ending was –osio (VALESIOSIO in Lapis Satricanus and perhaps TITOIO, see chapter 4 in Bakkum 2009 for a more detailed discussion).
In Old Latin (3rd-2nd centuries BCE), the genitive singular ending was only –ī, from *-ihx (cf. Leumann “im Altlatein nur –ī, nicht –iī”). This ending was contracted from -iī. This contracted form can be found, for instance, in Plautus (e.g. beneficī). Note this ending is thematic. Also, in masculine nouns the contraction did not affect stress position, thus gen.sg. Valéri.
Penny 2011 writes that "the Latin evidence does not allow more than speculation about the original function and distribution of the two endings" [i.e. -osio and -ī, Alex B.].
Later, due to analogical leveling, it was restored, first in adjectives and later in nouns; so, in Classical Latin (1st century BCE- 3rd/4th centuries CE), it was –iī as well as -i (cf. Leumann “In späterer Zeit aber gilt –iī als die richtiger Form” though).
Such forms are found in Andronicus (e.g. Saturni, uerbi), Vergil (e.g. fluviī), Propertius (imperiī, ingeniī) etc.
Leumann also mentions that "Das scheinbar regelmäßige -iī bei io-Stämmen, das schon Varro empfiehlt" and Varro lived in 116-27 BC. Tronskii 1960 adds that the -ī form was still used in the so called post-Augustan Latin (Silver Latin); he argues that Horace (i.e. Augustan) and Persius (post-Augustan) don't use the -iī genitive singular. So, there was a lot of variation.
Even in Augustus' Res Gestae (1: 22) we can find "magister conlegii" etc. The point is that all these changes did not happen overnight. Obviously, there was no decree making all Latin writers use the -iī after Augustus death.
Interestingly, re: auxilii - and the reliability of Latin corpora brought up here in some comments, Leumann remarks that
“Für Cicero und Caesar (auxilii Gall. 1, 31, 14 usw.) ist auf die handschriftliche Überlieferung kein Verlass” - basically, we don't really know which form they used (at least, with poetry there is metrical evidence).
I will add results of my corpus research later. For instance, I searched LASLA for Gr.Cat: Noun, Gr.Sub.Cat: 2nd decl., Case: Genitive, Number: Singular and it returned 16438 results - which I have to analyze manually now.