In order to answer to your question, I can suggest some things (no order in it !) :
*For single note improvisation, I suggest you to transcribe and/or play some Kurt Rosenwinkel (If Kurt was an interval, I think he would be a fourth !) and Mike Moreno lines (he plays a lot of modern licks using fourth based on the pentatonic). For Mike Moreno, you can find a transcription here of a solo on Isotope : http://www.shawnpurcell.com/products.html
You'll find on Ted Greene's Jazz Guitar Single Note Soloing, Volume 1 some great quartal exemples with fingerings.
Try using different fingers for perfect quartal single note lines. It helps sometimes for velocity and precision (I'm in this kind of fingering problems and I did not resolve it !). Playing a 4th on one string too, sometimes (1rst and 4th fingers !).
Take a look at Bergonzi's method (the one about pentatonics).
*Listen to pianists (begining with McCoy Tyner). If you can, transcribe their comp. If you can't, try to find piano transcriptions (adjust voicings for guitar : drop some notes or leave them when you need it) and play it and/or analyze it. Read all the jazz piano methods you can.
*Take a look at Mel Bay's Complete Book of Harmony, Theory & Voicing by Bret Willmott, especially on the Fourth Voicing Chapter (+ you've got there an harmonized pentatonic scale that can be used as a system. You can easily figure it if you take a quartal voicing with the low note on C and then move each note of the voicing to the next note of the C minor pentatonic and then move the same way for the next voicings... If you know what minor or major pentatonics you can play on every chords you see, you'll be able to use this harmonized scale in order to create interesting comp or play some chord melodies in a new way and with a little work, it will seem natural).
*You'll have to play harmonized scales in fourth for each kind of chords. And work on them in all tonalities and all string groupings (A, D, G, B and D, G, B, E are the grouping I use most of the time). Maybe play then mixing the string groupings in order to stay in the same area on the neck. Then, take a progression (working each kind of chord on the circle of 4th and 5th will help a lot I think) with chord changing every measure. And play 2 quartal voicings on the first chord moving up diatonicaly, begining as low as you think it can sound with a bass played under it, then play the 2 next voicings that feature the second chord and on... You'll have some smooth lines going up. When you think chords are played so high that it doesn't sound great (or if you can't go higher), change the direction... After that, you can make some variations : when the 2nd voicing of a measure is a voicing that can be played on the second chord, stay on this voicing for the first part of next measure and then go up... or play it using 3rds movement in the harmonized scales in fourth etc... That's just exercices. You'll have, in order to master it, to take every kind of chords and play and find the quartal chords that feature the best this chord. For exemple, take a II-7 chord (with a function of subdominant, "dorian" if you want). Play the harmonized "dorian" scale in fourth on it and see what notes each fourth chord is composed of, and his interval with the Root (R). And listen to his sound with the bass. You'll notice that some sounds great in tonal context, other not, some poor, other not. And then make a hierarchy with them.
Chord I : R, 4, b7, m3
Chord II : 9 , 5, R, 4
Chord III : m3, 6, 9, 5
Chord IV : 4, b7, m3, 13
Chord V : 5, R, 4, 7
Chord VI : 6, 9, 5, R
Chord VII : b7, m3, 13, 9
In order to create a hierarchy you'll have to make choices. For my own choices, I first leaved all chords that contains the 6th (or 13). The more "important" chord is the one that contains both the m3 & the b7, so the Chord I is the more important. Then the Chord V, and the Chord II. The other chords can be played as passing chords. Or/and you'll have to make them less important in the measure (because of duration, of situation, accentuation, or other things... use them as approach chords, neighbor chords etc... , or passing chords between the "important" chords).
Then do an equivalent work with other kind of chords.
For exemple, for IM7 (Tonic, Ionian) and for V7 (dominant, "mixolydien"), leave the 4th, for VII-75b (subdominant in a minor context, Locrien), leave the b9 etc... For the chords derived from the Major scale, there is always a note that you can, in tonal context, consider as an "avoid" note, excepted for the IVM7 ("Lydian"). On this last one you can play each quartal voicing because there is no "avoid" note (and it helps a lot !). If you use the dorian for a I-7 (Tonic, dorian), this is the same (idem... !). It is the same too for each chord derived of the melodic scale (idem !)...
In a modal context, you'll have to change this concepts, but I suppose you wanted to have answers about quartal voicings in a tonal context.
I must add that in order to play smoother lines, you can add some chromatic passing chords. And I think you'll have to mix quartal voicings with other voicings...
Another thing you can do is to adapt voicings that contain the "avoid note". For exemple, you can move the "avoid" note of the voicings it appears on. The 6th here can become a 5th or a b7. Then you'll have a chord scale with quartal voicings and adaptations of quartal voicings.
So you'll have, if we stay with that same II-7 exemple :
Chord I : R, 4, b7, m3
Chord II : 9 , 5, R, 4
Chord III : m3, b7 (because a 5th would produce an octave interval) , 9, 5
Chord IV : 4, b7, m3, 5 (because a 7th would produce an octave interval)
Chord V : 5, R, 4, b7
Chord VI : b7 (...), 9, 5, R
Chord VII : b7, m3, 5 (...), 9
*You can use constant voicing structure (maybe I'll develop later) with each kind of quartal voicing.
*You maybe should begin exploring 3 notes (triadic) quartal harmony first. It is more versatile and flexible. And maybe the 2 inversions of them (you can name them "sus2" : Root, 2nd, 5th, and "sus4" chords : Root, 4th, 5, for the inversions of the perfect quartal triad, or "7sus4omit5" : Root, 4th, 7th) : I hear a lot of this 2 inversions in Kurt Rosenwinkel's playing (If Kurt was a chord...). Play harmonized scales too with them...
I'll add some tab exemples later on begining of Autumn Leaves.
I hope that it helps a little and that it is not confused,
best regards.
