Prior to Husserl, Brentano discusses the immortality of the soul in the first chapter on "The Concept and Purpose of Psychology" of his 1874 book Psychology from an Empirical Standpoint. Despite the fact that psychology would no longer be "the science of the soul" but rather "the science of mental phenomena", it would still be capable of addressing the idea of immortality of "mental life", because there is a continuity of mental life that does not require the assumption of a substantial soul (which inner perception does not show us). The laws of psychology would still hold after our bodily death:
"When we depart from this life we separate ourselves from all that is
subject to the laws of natural science. The laws of gravitation, of
sound, of light and electricity disappear along with the phenomena for
which experience has established them. Mental laws, on the other hand,
hold true for our life to come as they do in our present life, insofar
as this life is immortal."
This would seem to fit a "phenomenological" approach to immortality, though not a Husserlian one.