A running tool with an overshot sleeve is provided. In one embodiment, a system includes a running tool and an overshot sleeve including a first portion threaded onto the running tool and a second portion. The first portion and the second portion are connected together in a manner that allows the first portion and the second portion to rotate with respect to one another. The second portion of the sleeve can engage an additional component, such as a casing hanger, and the running tool and the overshot sleeve can be configured to selectively convert rotation of the running tool into axial motion of the overshot sleeve with respect to the running tool. Additional systems, devices, and methods are also disclosed.