This work proposes a new term on the "Jacob's ladder" of approximate exchange-correlation functionals in Kohn-Sham density functional theory. In these Rung 3.5 functionals, the exchange-correlation energy density at a point depends linearly (rather than quadratically) on the nonlocal Kohn-Sham one-particle density matrix in a finite neighborhood around the point. These functionals are intermediate in complexity between the semilocal approximations of Rungs 1-3, and fully nonlocal Rung 4 approximations such as global hybrids. Rung 3.5 functionals built on the model for exchange in [B. G. Janesko, J. Chem. Phys. 131, 234111 (2009)] predict molecular thermochemistry and kinetics with accuracy intermediate between their "parent" semilocal functional and the corresponding global hybrid. The best Rung 3.5 functional presented here gives mean absolute errors of 5.7 kcal/mol for G3/99 thermochemistry, 5.2 kcal/mol for HTBH38/04 hydrogen-transfer reaction barriers, and 5.7 kcal/mol for NHTBH38/04 nonhydrogen-transfer reaction barriers, while incorporating only two empirical parameters.