The Twins placed Smith on the 15-day injured list Sunday with upper trapezius tightness.
Smith's IL stint is retroactive to Friday, so he won't be eligible for activation until July 9. The 38-year-old has been dependable in a setup role for the Twins in 2022, collecting a win and 14 holds while supplying a 2.78 ERA and 1.46 WHIP across 22.2 innings.