The author reconsiders the oblate spheroidal functions satisfying the differential equation
for the case where , , with and , and gives their asymptotic expansions for while . The expansions are uniformly valid in certain subdomains of the -plane [Notice that the case was already studied by the author himself, ibid. 121, No. 3/4, 303-320 (1992; Zbl 0781.33010)]. Explicit error bounds are given for all the approximations. Some asymptotic behaviour of spheroidal functions in terms of elementary functions of complex arguments are also derived. These latter are used to determine relationships between the parameters , , and (the characteristic exponent).