This paper deals with an interpolation problem in the open unit disc $𝔻$ of the complex plane. We characterize the sequences in a Stolz angle of $𝔻$, verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on $𝔻$, but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.