  Steiner quadruple systems are set systems in which every triple is contained
in a unique quadruple. It is will known that Steiner quadruple systems of order
v, or SQS(v), exist if and only if v = 2, 4 mod 6. Universal cycles, introduced
by Chung, Diaconis, and Graham in 1992, are a type of cyclic Gray code. Overlap
cycles are generalizations of universal cycles that were introduced in 2010 by
Godbole. Using Hanani's SQS constructions, we show that for every v = 2, 4 mod
6 with v > 4 there exists an SQS(v) that admits a 1-overlap cycle.
