This paper proposes a post-model selection inference procedure, called targeted undersmoothing, designed to construct uniformly valid confidence sets for a broad class of functionals of sparse high-dimensional statistical models. These include dense functionals, which may potentially depend on all elements of an unknown high-dimensional parameter. The proposed confidence sets are based on an initially selected model and two additionally selected models, an upper model and a lower model, which enlarge the initially selected model. We illustrate application of the procedure in two empirical examples. The first example considers estimation of heterogeneous treatment effects using data from the Job Training Partnership Act of 1982, and the second example looks at estimating profitability from a mailing strategy based on estimated heterogeneous treatment effects in a direct mail marketing campaign. We also provide evidence on the finite sample performance of the proposed targeted undersmoothing procedure through a series of simulation experiments.