The 6th Texas Infantry had two known flags at least.
The one you have been discussing is their First National flag that bears a blue leather glove in the white bar. The second was a red merino flag with a blue shield in the center bearing 13 stars (12 along each side of the shield and a larger star in the center) and the unit name along the bottom.
Sadly, the history of the making of these flags seems to be identical which means that, 1) either one of those is wrong or, 2) both flags were made by the same ladies.
The First National with the glove bears the name "Kate Phillips" on the reverse side of the glove. She is supposedly the one that made this flag - not Mrs. Richard Owens and her daughters. Mrs. Owens is tied to making the red flag with blue shield - which is missing today.
Both flags were captured at Arkansas Post, AR in January 1863. The First National was taken by a soldier of the 15th Missouri who evidently moved to Pennsylvania after the war and became a member of the GAR Hall in Williamsport, PA. The flag went from the GAR hall to the local historical society who then returned it to Hill College in Texas in 1991.
I have seen the First National flag with the glove several times over the years. It is on display in the Texas Heritage Museum at Hill College which is south of the Dallas-Fort Worth Metroplex about 40 miles. A good museum with several flags on display.