"Ger Yihyeh Zar'echa Ba'eretz Lo Lahem" - As the Jewish people are considered to be quintessential wanderers, Chazal state that it is vitally important to keep your towel. This was especially important in the midbar, as the towel could provide shade, allow you to comfortably lay upon the sand, or, as in Moshe's case, double as a mask to hide his glowing face from the nation.
Chazal are not referring to a chain of blankets, they are referring to a specific CHAIN OF OWNERSHIP over a SINGLE TOWEL, specifically the heilege towel Hashem handed Moshe rabbeinu to clean up after writing the first Sefer Torah. This towel obviously doubled as the Masveh, which is in fact the end-shortening of the word Mesorah, where the Reish and Heh are combined into a single letter.
The original process of conferring smicha involved wrapping the towel around the Talmid's head and THEN laying your hands upon the towel turban. The Mitznefet of the Kohen Gadol was worn with the tzitz to symbolize exactly this divine process. This is why the KG cannot raise his hands above the Tzitz, as it would would be placing his hands on top of Hashem's hand, something explicitly forbidden since Chazal say that Hashem purchased Yisroel and therefore "Yado al ha'elyonah."
Theodosius passed laws forbidding towel transmission, leading to the original towel being hidden and it's location lost. It is assumed they passed this decree because the Romans believed the proliferation of towel use among Jews detracted from the glory of their aqueducts and bathhouses, and that forbidding the use of towels was an attempt to enforce assimilation of the Jews into the culture of the Roman Empire.
There is significant dispute among the Sages as to whether a new towel can be ordained with the holiness necessary to reinstate the process of the rabbinic mesorah. Related debates rage about whether the composition of the towel is important, particularly whether the towel should be composed like the mitznefet or the paroches and whether shatnez is a halachic concern in the construction of a new S'michah.