We present results of tensor-network simulations of the three-dimensional $O(2)$ model at non-zero chemical potential and temperature, which were computed using the higher-order tensor-renormalization-group method (HOTRG). This necessitated enhancements to the HOTRG blocking procedure to reduce the truncation error in the case of anisotropic tensors. Moreover, the construction of the truncated vector spaces was adapted to strongly reduce the effect of systematic errors in the computation of observables using the finite-difference method. Our (improved) HOTRG results for the evolution of the number density with the chemical potential are in agreement with results obtained with the worm algorithm, and both the Silver Blaze phenomenon at zero temperature and the temperature dependence of the number density can be adequately reproduced.