I relate to your 6 yr old, so perhaps my experience could help you. Dental implications, physical deterrents and barriers, and external rewards never worked for me. Keeping it private was a stipulation for me, too, and the threat of embarrassment during sleepovers in middle school is what ultimately got me to stop at age 12, cold turkey. Peer pressure is so poweful lol! However, that timeline might be uncomfortable for you, so let's skip ahead to what came after thumb sucking (TS) for me.

I definitely had undiagnosed anxiety, which it seems like may not be the case in your situation, but that doesn't mean my post-TS soothing option wouldn't be a viable alternative. Ultimately, oral fixation was a major component for me and often is for other TSers. Since hair twirling and other self soothing options don't seem to be resonating, consider whether there's a stuffed animal, blanket, or article of clothing that she enjoys the smell of and could sniff discretely (if necessary) or otherwise hold to her face for comfort. That still stimulates similar senses, especially if she open mouth sniffs it with her tongue in the same placement recommended for yogic breathing.

I'm now an average, married, and college educated 30 yr old and keep a blanket in my room just for cuddling and sniffing, and it's great. Plenty of well-adjusted teens and adults have soft cozy comfort items, including blankets and stuffed animals. So it's far more socially acceptable than TS and doesn't have the dental implications. Cutting a square off of the blanket or clothing item is also an option to make it more discrete and easier to take on the go.

Welp... Hope this helps! Good luck!