Advection of trace species, or tracers, also called tracer transport, in models of the atmosphere and other physical domains is an important and potentially computationally expensive part of a model's dynamical core. Semi-Lagrangian (SL) advection methods are efficient because they permit a time step much larger than the advective stability limit for explicit Eulerian methods without requiring the solution of a globally coupled system of equations as implicit Eulerian methods do. Thus, to reduce the computational expense of tracer transport, dynamical cores often use SL methods to advect tracers. The class of interpolation semi-Lagrangian (ISL) methods contains potentially extremely efficient SL methods. However, unstructured grids, as E3SM uses, are a challenge for ISL methods. We developed an element-based ISL transport method that we call the Interpolation Semi-Lagrangian Element-based Transport (Islet) method, such as for use with atmosphere models discretized using unstructured element-based methods, including the E3SM Atmosphere Model. A core piece of the Islet method is a collection of element basis sets that provide spatial discretization. In contrast to classical ISL methods, the Islet method works on unstructured element grids. The natural element basis sets lead to an unstable ISL method. We developed a procedure to search for modified, stabilized basis sets. The resulting basis sets provide extremely accurate tracer transport and excellent diagnostic values in a number of verification problems. The lowest-order basis set is used in the E3SM Atmosphere Model versions 2 and 3 and is part of the overall transport method that speeds up atmosphere tracer transport over version 1 by a factor of six to eight.