At least in the case of "1960's soul to 1990's house," I share Colin Fine's view that you don't need an apostrophe at all. Some style guides undoubtedly differ on this matter of punctuation—as they do on many others—but a number of major style guides the oppose using an apostrophe before the s in decades or centuries rendered as numerals.
From The Associated Press Stylebook (2007):
years ... Use an s without an apostrophe to indicate the span of decades or centuries: the 1890s, the 1800s.
From The Chicago Manual of Style, sixteenth edition (2010):
9.34 Decades. Decades are either spelled out (as long as the century is clear) and lowercased or expressed in numerals. Chicago calls for no apostrophe to appear between the year and the s. [Examples:] the nineties [and] the 1980s and 1990s (or, less formally, the 1980s and '90s)
From The Oxford Guide to Style (2002):
4.1.6 Geological periods and events ... To denote simple ten-year spans OUP style prefers, for example, 1920s or 1960s to nineteen-twenties or nineteen-sixties. To denote decades of a specific character (say the Roaring Twenties or the Swinging Sixties) OUP prefers Twenties or Sixties to '20s or '60s.
So, depending on which of these three style guides you happened to be following, you would render your example phrase as
1960s soul to 1990s house [AP style]
sixties soul to nineties house or 1960s soul to 1990s house or, less formally, '60s soul to '90s house [Chicago style]
1960s soul to 1990s house [Oxford style]
And even if you choose what Chicago calls the less formal approach of shortening decades with apostrophes, there is no ambiguity in the result:
'60s soul to '90s house
Of course, it is possible to construct a phrase where a true possessive form of a decade is necessary, as in
the '90s' main contribution to popular culture
but all three style guides cited above would enable you to minimize the possibility of a misreading of the punctuation by using the form
the 1990s' main contribution to popular culture
If you refuse to abandon your personal preference for '90s, the only way you can avoid subjecting your readers to needlessly ambiguous punctuation is to recast the phrase:
the main contribution of the '90s to popular culture