There are a lot of ways to think about this, in part because quartal voicings are so ambiguous. It opens the door to many different scale and chord pairings.
Harmonize a Melody Note with a Quartal Voicing Outside the Key
This is something we can hear Chick Corea doing. Imagine we're improvising over a Cmin chord and we want to play a lick ending on E♭. Instead of harmonizing the E♭ with a traditional Cmin chord, we could play the quartal voicing D♭-G♭-B♮ in our left hand, along with the E♭ in our right hand. The left hand thus "goes out," but the entire voicing D♭-G♭-B♮-E♭ sounds good.
This is something valuable to practice. Pick a note at random (like E♭), and in the left hand, harmonize the note with different quartal voicings that sound good. (It's worth noting that quartal voicings can be inverted.) For example, some quartal voicings that I like under E♭ are:
Use Movement of the Quartal Chords to Go Out
Another way to go out is to let the left hand take the lead. That is, play a quartal voicing that fits in the designated key. Then move the quartal voicing up or down a half step, a minor third, or any other interval which introduces notes that are outside the designated key.
Once you have your quartal voicing, pick a scale that works with it. An easy way to do this is to build a pentatonic scale from any one of the three notes in the quartal chord. For the voicing F-B♭-E♭, the "best fitting" pentatonic scales are
F minor pentatonic,
B♭ minor pentatonic, and
E♭ Major pentatonic.
Or instead of using a pentatonic scale, another quick trick is to use a triad pair.1 Build the triad pair from the top note of the quartal voicing (E♭Maj triad) and a whole step down (D♭Maj triad). So over the quartal voicing F-B♭-E♭, we could use the triads E♭Maj and D♭Maj. This is another great way to practice--pick a quartal voicing at random, and then improvise using the associated triad pairs.
Go Out the Traditional Way
Another more traditional approach is to simply pick a new scale that deviates from the given scale. For example, if you're playing over a Cmin chord, you could start playing in D♭min and then pull quartal voicings from the D♭min scale. Using this technique, common intervals to use are a half step, a minor third, and a tritone. So if you're improvising over a Cmin chord, you might switch to Bmin or D♭min, Amin or E♭min, or G♭min.
1. Generally, this technique involves building a six-note scale from two triads. Soloing built from this technique often use arpeggios that alternate between the two triads. There are good videos on YouTube about this technique.