A Blaschke product $B$ with zero-sequence $(a_n)$ is called almost interpolating if the inequality $\liminf_n(1-|a_n|^2)|B'(a_n)|\geqslant \d>0$ holds. The sets $U$ for which there exists a Blaschke product $B$ such that $(a-B)/(1-\ov a B)$ is almost interpolating if and only if $a \in U$ are studied. Examples of such sets include open sets, containing the origin, and whose complement is the closure of an arbitrary set of concentric open arcs around the origin or open sets whose complement is of zero logarithmic capacity. Results on the range of interpolating Blaschke product s on the set of trivial points in the spectrum of $H^{\infty}$ are deduced.
