Multiple coverings of the farthest-off points ((R, µ)-MCF codes) and the corresponding (ρ, µ)-saturating sets in projective spaces P G(N, q) are considered. We propose and develop some methods which allow us to obtain new small (1, µ)-saturating sets and short (2, µ)-MCF codes with µ-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly 1 + 1/cq, c ≥ 1). In particular, we provide new algebraic constructions and some bounds. Also, we classify minimal and optimal (1, µ)-saturating sets in P G(2, q), q small.