Let M be a module. A delta-cover of M is an epimorphism from a module F onto M with a delta-small kernel. A delta-cover is said to be a flat delta-cover in case F is a flat module. In the present paper, we investigate some properties of (flat) delta-covers and flat modules having a projective delta-cover. Moreover, we study rings over which every module has a flat delta-cover and call them right generalized delta-perfect rings. We also give some characterizations of delta-semiperfect and delta-perfect rings in terms of locally (finitely, quasi-, direct-) projective delta-covers and flat delta-covers.