  We develop the derivative expansion of the one-loop ${\cal N}=4$ SYM
effective action depending both on ${\cal N}=2$ vector multiplet and on
hypermultiplet background fields. We get a new derivation of the complete
${\cal N}=4$ supersymmetric low-energy effective action obtained in
hep-th/0111062 and find subleading corrections to it. A problem of ${\cal N}=4$
supersymmetry of the results is discussed. Using the formalism of ${\cal N}=2$
harmonic superspace and exploring on-shell hidden ${\cal N}=2$ supersymmetry of
${\cal N}=4$ SYM theory we construct the appropriate hypermultiplet-depending
contributions. The hidden ${\cal N}=2$ supersymmetry requirements allow to get
a leading, in hypermultiplet derivatives, part of the correct ${\cal N}=4$
supersymmetric functional containing $F^{8}$ among the component fields.
