To supplement whatever you interested folks might have encountered in the Stanford Encyclopedia of Philosophy and the site for the “Center for Advanced Research in Phenomenology,” run by Lester Embree (a student of students of Husserl).
This site pulls text from Embree’s introduction to the very overpriced Encyclopedia of Phenomenology (only $789!) to characterize some “widely accepted features of the phenomenological approach” (the last of which is, amusingly to me, that “phenomenologists tend to debate whether or not what Husserl calls the transcendental phenomenological epochê and reduction is useful or even possible.”).
What interested me here primarily was Embree’s breakdown of “tendencies and stages within philosophical phenomenology,” which tells you how Husserl fits in with later folks:
(1) Realistic phenomenology emphasizes the search for the universal essences of various sorts of matters, including human actions, motives, and selves. Within this tendency, Adolf Reinach added philosophy of law to the phenomenological agenda; Max Scheler added ethics, value theory, religion, and philosophical anthropology; Edith Stein added philosophy of the human sciences and has been recently recognized for work on gender; and Roman Ingarden added aesthetics, architecture, music, literature, and film…. This tendency… flourished in Germany through the 1920s, but also continues today.
(2) Constitutive phenomenology’s founding text is Husserl’s ["Ideas"] of 1913. This work extends Husserl’s scope to include philosophy of the natural sciences, which has been continued in later generations by Oskar Becker, Aron Gurwitsch, and Elisabeth Ströker, but it is chiefly devoted to reflections on phenomenological method, above all the method of transcendental phenomenological epochê and reduction.
This procedure involves suspending acceptance of the pregiven status of conscious life as something that exists in the world and is performed in order to secure an ultimate intersubjective grounding for the world and the positive sciences of it. Use of this method places constitutive phenomenology in the modern tradition that goes back at least to Kant, and also characterizes the rest of Husserl’s work…
(3) Existential phenomenology is often traced back to Martin Heidegger’s [Being and Time] of 1927, the project of which was actually to use an analysis of human being as a means to a fundamental ontology that went beyond the regional ontologies described by Husserl.
Hannah Arendt… seems to have been the first existential phenomenologist after Heidegger. It is also arguable that existentialist phenomenology appeared in Japan with Miki Kyoshi and Kuki Shuzou’s early work in the late twenties. However, this third aspect and phase in the tradition of the movement took place chiefly in France. The early Emmanuel Levinas interpreted Husserl and Heidegger together and helped introduce phenomenology into France. This period included Gabriel Marcel and was led in the 1940s and 1950s by Simone de Beauvoir, Maurice Merleau-Ponty, and Jean-Paul Sartre.
This third tendency is concerned with topics such as action, conflict, desire, finitude, oppression, and death. Arendt contributed to political theory and the problematics of ethnicity, Beauvoir raised the issue of gender and old age, Merleau-Ponty creatively continued the appropriation of Gestalt psychology in his descriptions of perception and the lived body, and Sartre focused on freedom and literature…
(4) Hermeneutical phenomenology chiefly stems from the method set forth in Heidegger’s [Being and Time], according to which human existence is interpretative. The first manifestation of this fourth tendency is Hans-Georg Gadamer… The issues addressed in hermeneutical phenomenology include simply all of those that were added to the agenda in the previous tendencies and stages. What is different is the emphasis on hermeneutics or the method of interpretation. This tendency has also included much scholarship on the history of philosophy and has had extensive influence on the human sciences.
While realistic and constitutive phenomenology arose and first flourished in Germany before and after World War I and existential phenomenology spread out from France after World War II, hermeneutical phenomenology appears to have been most actively pursued in the United States during the 1970s and 1980s.
At the very least, this gives you some additional terms and names on which you can search to do more research. It should be clear that, contra the impression I may have given on our Husserl episode, phenomenology is neither some idiosyncratic thing that began and ended with Husserl or is so diffuse that any sort of self-reporting that goes on in psychological research counts as phenomenology in the philosophical sense. I’ve commented that Dan Dennett, for instance, is all for throwing out our Cartesian reliance on introspection as a source of legitimate knowledge, but in Dennett’s article we read for our philosophy of mind episode, he actually uses some kind of phenomenological method to argue against the prima facie results of introspection. Simply “describing experience” isn’t going to do it: in Husserl’s words you have to get at experience’s “structures,” which means, in the Dennett case (you can search for “osprey cry” in this article to find exactly what I’m talking about), having repeated experiences of the same apparent object under different circumstances is necessary to adequately get at the phenomena, though certainly our first impression and how it evolves with more information is part of the structure of that overall chain of experience.