The answer is catagen. Milady describes this transition phase in language that matches the question exactly: during catagen, the follicle shrinks, the hair bulb disappears, and the shrunken root end forms a rounded club.
The book notes catagen lasts only 1 to 2 weeks, with less than 1 percent of scalp hair in this phase at any given time — genuinely the shortest of the three named phases. Anagen (growth) comes before it and telogen (resting) comes after, making this the brief middle transition between the two.
Milady Standard Barbering, 6th ed., Ch. 10 (Properties and Disorders of the Hair and Scalp), p. 256: "During the catagen phase the follicle shrinks, the hair bulb disappears, and the shrunken root end forms a rounded club."
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