The state Supreme Court's order comes one day after a state appeals court had allowed the expansion to stand while a legal case was appealed.
Early voting for the July 14 runoffs will begin June 29, instead of July 6, under a proclamation issued Monday by Gov. Greg Abbott.
In their suit, a group of Texas voters and civic organizations claim that four specific Texas absentee voting rules are unconstitutional during a public health crisis.
The voters — all between the ages of 18 and 28 — want the courts to rule that the state's age restriction for voting-by-mail, which limits eligibility to those 65 and older, violates constitutional protections.
A state district judge on Wednesday said he will issue a temporary injunction expanding who can qualify for an absentee ballot for the upcoming elections. The decision faces certain appeal as Democrats and Republicans clash over voter access.