CHARLESTON, S.C. — It was 10 a.m., less than 12 hours after the final performance of the Vivaldi rarity “Farnace” at the Spoleto Festival USA here, and Anthony Roth Costanzo’s facial hair was already a thing of the past.

The “Farnace” production, set in a vaguely modern Middle East, featured the much-in-demand countertenor Mr. Costanzo, 35, as a sort of rebel, complete with what he called his “Aleppo beard.”

But Mr. Costanzo was clean-shaven and up bright and early on this June morning to unveil a vehicle for his plush yet precise voice: a Nashville-tinged art-song reorchestration of Roy Orbison’s “Crying.” Later that afternoon, he would delight an audience at Spoleto’s chamber music series with boyhood tales of singing backup for Michael Jackson and the Olsen twins before donning Orbisonian sunglasses and singing that timeless weepy.

“Music written before 1750 or after 1950,” Mr. Costanzo said after the performance. “That’s my life.”