This has probably been covered a zillion times but I just wanted to say how I deal with it.
Firstly I rationalize chords into major, minor and dominant flavours. In major I include all Major,Major 7th,Augmented Maj 7th,Augmented,Major 7th #11,,M6,M9, any Add chords etc etc.Any chord with a major third with or without a major 7th. Minor chords include minor,minor 7ths,minor major 7ths,m6ths etc etc. Chords having a minor 3rd with or without a minor 7th. In the dominant camp I throw the dominant 5th, Dom 7th, Aug 7ths, any other altered chords,m7th flat 5s,Dim 7ths.Basically chords a major 3rd and minor 7th or a minor 3rd and a diminished fifth. Tritone intervals baby between root and flat 5. Suspended chords I treat as minor or major and power chords. The same.
So now I have my flavours what do I do with only 3 strings when confronted with a chord needing four or more.Well this is what I do.
For anything major with more than four I either just play a basic 1 3 5 triad or I chop out any notes other than than the major 3rd.For example say an Bbmaj7th #11,I will ditch the 5th and the 7th and play 1 3 #11. even though its got a tritone interval it still sounds like a resting chord due to the major 3rd. What about a Dm major 7th.Well I get rid of the 5th and hit the 1 m3 M7.
Getting into dominant territory now. Dom 7ths I ditch the 5th. Altered chords like a C7#9 I choose the 1 m7 # 9. For dim 7ths I go with 1 m3 flat 5.
I know none of this is news to a lot of guitar players.I guess what I am trying to say is that when encountering chords with more than 3 notes you gotta prioritize what gives the chord its special flavour.