I usually cheat that chord by not fretting the 6th string 1st fret but play everything else the same. Unless you really need that lower tone I usually skip it if I'm playing busy chord progressions.
This song for example; which uses the F maj chord in the verse, I usually skip the lower tones in the F until the turn around, where the progression repeats, where you play only a partial F maj - 6th string to 4th.