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I switched my e-ring to my right ring finger for the ceremony so my band was first, then switched it after.
If you chose to do this route, my advice is to let your groom know you are doing it! H tried to put the band on my right hand because that is where he saw my e-ring.
I didn't wear mine during the ceremony. H put it back on when he put my band on. I don't think it matters either way though.