Fundamentally, the core purpose of silk is to be be made into clothing (or later, writing material, and perhaps bedding). In this sense, what is probably the earliest silk remains, dating to about 2700 B.C., were excavated from the Qianshanyang Historical Site
(錢山漾遺址) in modern Wuxing, Zheijang. At least some of those silk would have been made to be worn.
In terms of actual writing on silk found, you already have your answer. The Chu Silk Manuscript you linked dates to 300 B.C., and is the earliest known silk manuscript. There are some other documents found together with it but AFAIK those were never published in their entirety. Most excavated documents from this era (Warring States, 475-221 B.C.) comes in the form of bamboo and wooden slips. Silk in general were not well preserved.
That manuscript is actually series of drawings surrounding some annotating text. Two other silk paintings are known to exist from the Warring States period, and may slightly pre-date the Chu Silk Manuscript:
- The Lady, the Dragon and the Phoenix
- The Man Riding a Dragon
The cache at Mawangdui is the earliest major silk manuscript find in Chinese archaeological history.