This paper extends many conclusions based on phantom envelopes and Ext-phantom covers of modules, and we find that many important properties still hold after replacing phantom and Ext-phantom with n-phantom and n-Ext-phantom respectively. In addition, we also obtain some extra results. Specifically, we give a characterization of the weak dimensions of rings in terms of n-phantom envelopes and n-Ext-phantom covers of modules with the unique mapping property respectively. We show that wD(R) ≤ 2n whenever every right R-module has an n-phantom envelope with the unique mapping property or every left R-module has an n-Ext-phantom cover with the unique mapping property over left coherent rings.