I research the history of the 26th Illinois, and in all of the information I have found in the past 10 years of digging through primary and secondary source, I've never heard of the 26th losing a flag during their service.

Similarly, as I understand it anyway, Loomis' Brigade was pinned down by fire the duration of their time in front of Tunnel Hill. If any of Loomis' other regiments lost their flags, I'm not aware of it.

Hope this helps,
Brad