You should consult Taittiriya Upanishad part 2 which talks about the five Koshas surrounding the Atman. The five koshas are annamaya, pranamaya, manomaya, Vijnanamaya and Anandamaya Koshas. These 5 koshas or sheaths are organised into two bodies the Sthula sarira or the gross physical body and the sukshma or linga sarira also known as the subtle body. It is the subtle body that survives death and is reborn. For example, Gita 15.8 says:
When he gets a new body or abandons an old one, the Jiva, the lord of
the body, moves, carrying them (the mind and the senses) with him, as
the wind carries smells from their seats (in flowers and the like).
The mind and the senses form the subtle body. The subtle body dies only after attainment of moksha. A description of Moksha which explains this is given in Mahabharata:
When men of knowledge, conversant with the rules of Yoga, become as
fixed as a stake of wood, and as immovable as a mountain, then are
they said to be in Yoga. When one does not hear, and smell, and taste,
and see; when one is not conscious of any touch; when one’s mind
becomes perfectly free from every purpose; when one is not conscious
of anything, when one cherishes no thought; when one becomes like a
piece of wood then is one called by the wise to be in perfect Yoga.
At such a time one shines like a lamp that burns in a place where
there is no wind; at such a time one becomes freed even from one’s
subtle form, and perfectly united with Brahma. When one attains to
such progress, one has no longer to ascend or to fall among
intermediate beings. When persons like ourselves say that there has
been a complete identification of the Knower, the Known, and
Knowledge, then is the Yogin said to behold the Supreme Soul.
Mahabharata Santi Parva Section CCCVII
