Aims. Previous radio recombination line (RRL) observations of dust clumps identified in the APEX Telescope Large Area Survey of the Galaxy (ATLASGAL) have led to the detection of a large number of RRLs in the 3 mm range. Here, we aim to study their excitation with shorter wavelength (sub)millimeter radio recombination line (submm-RRL) observations. Methods. We made observations of submm-RRLs with low principal quantum numbers (n ≤ 30) using the APEX 12 m telescope, toward 104 H II regions associated with massive dust clumps from ATLASGAL. The observations covered the H25α, H28α, and H35β transitions. Toward a small subsample the H26α, H27α, H29α, and H30α lines were observed to avoid contamination by molecular lines at adjacent frequencies. Results. We have detected submm-RRLs (signal-to-noise (S∕N)≥ 3 σ) from compact H II regions embedded within 93 clumps. The submm-RRLs are approximately a factor of two brighter than the mm-RRLs and consistent with optically thin emission in local thermodynamic equilibrium (LTE). The average ratio (0.31) of the measured H35β/H28α fluxes is close to the LTE value of 0.28. No indication of RRL maser emission has been found. The Lyman photon flux, bolometric, and submm-RRL luminosities toward the submm-RRL detected sources present significant correlations. The trends of dust temperature and the ratio of bolometric luminosity to clump mass, L_bol ∕M_clump, indicate that the H II regions are related to the most massive and luminous clumps. By estimating the production rate of ionizing photons, Q, from the submm-RRL flux, we find that the Q(H28α) measurements provide estimates of the Lyman continuum photon flux consistent with those determined from 5 GHz radio continuum emission. Six RRL sources show line profiles that are a combination of a narrow and a broad Gaussian feature. The broad features are likely associated with high-velocity ionized flows. Conclusions. We have detected submm-RRLs toward 93 ATLASGAL clumps. Six RRL sources have high-velocity RRL components likely driven by high-velocity ionized flows. Their observed properties are consistent with thermal emission that correlates well with the Lyman continuum flux of the H II regions. The sample of H II regions with mm/submm-RRL detections probes, in our Galaxy, luminous clumps (L_bol > 10⁴ L_⊙) with high L_bol∕M_clump. We also provide suitable candidates for further studies of the morphology and kinematics of embedded, compact H II regions with the Atacama Large Millimeter/submillimeter Array (ALMA).