We present a photometric analysis of the star clusters Lindsay 1, Kron 3, NGC 339, NGC 416, Lindsay 38, and NGC 419 in the Small Magellanic Cloud (SMC), observed with the Hubble Space Telescope Advanced Camera for Surveys (ACS) in the F555W and F814W filters. Our color-magnitude diagrams (CMDs) extend ~3.5 mag deeper than the main-sequence turnoff points, deeper than any previous data. Cluster ages were derived using three different isochrone models: Padova, Teramo, and Dartmouth, which are all available in the ACS photometric system. Fitting observed ridgelines for each cluster, we provide a homogeneous and unique set of low-metallicity, single-age fiducial isochrones. The cluster CMDs are best approximated by the Dartmouth isochrones for all clusters, except for NGC 419 where the Padova isochrones provided the best fit. Using Dartmouth isochrones we derive ages of 7.5 ± 0.5 Gyr (Lindsay 1), 6.5 ± 0.5 Gyr (Kron 3), 6 ± 0.5 Gyr (NGC 339), 6 ± 0.5 Gyr (NGC 416), and 6.5 ± 0.5 Gyr (Lindsay 38). The CMD of NGC 419 shows several main-sequence turnoffs, which belong to the cluster and to the SMC field. We thus derive an age range of 1.2-1.6 Gyr for NGC 419. We confirm that the SMC contains several intermediate-age populous star clusters with ages unlike those of the Large Magellanic Cloud and the Milky Way. Interestingly, our intermediate-age star clusters have a metallicity spread of ~0.6 dex, which demonstrates that the SMC does not have a smooth, monotonic age-metallicity relation. We find an indication for centrally-concentrated blue straggler star candidates in NGC 416, while these are not present for the other clusters. Using the red clump magnitudes, we find that the closest cluster, NGC 419 (~50 kpc), and the farthest cluster, Lindsay 38 (~67 kpc), have a relative distance of ~17 kpc, which confirms the large depth of the SMC. The three oldest SMC clusters (NGC 121, Lindsay 1, and Kron 3) lie in the northwestern part of the SMC, while the youngest (NGC 419) is located near the SMC main body.