Nobody except Tama and Ramillies has given you the mechanical answer I think you're probably looking for. And neither Tama and Ramillies were as clear as they could have been. Let's break things down in more detail, using the box method you're familiar with. (Tama's recommendation about scale degrees is a good one, but it doesn't help you right now. This will.)
Let's say you're moving up the neck in a particular passage during your solo. You know ahead of time which is your lowest note and which is your highest note for that passage. Given that information, you first decide which box it makes the most sense to start out with and which box you want to end up in. (That could vary depending on the previous passage and the upcoming passage, but that's irrelevant to your actual question, so let's not get sidetracked.)
So your real question is: How do I move from box to box?
Tama and Ramillies each give a different answers to this, and Tama especially was unclear--her answer was almost more implied than clearly stated (she says "The fingering pattern would be 124, 124, 1 slide 124" without actually explaining what she means), so I'll spell it out. Fortunately, her diagrams do help. To do what she's talking about, you slide from position to position. Here's what that means:
- When you reach the note from which you begin the slide, you do not continue playing in the current box. Yes, you play that note with the correct finger for the box, just as you've practiced it when you learned that box. But not the next note. Instead of playing the next note the way you should for that box, which would be with a different finger, you use the same finger to move up the same string until you get to the next note in the solo (passage).
- Once you reach that subsequent note, play it with the same finger. That's what is meant by "slide". (You can also slur while you slide--i.e. keep your finger pressed down hard on the string so the string continues to sound as you move--but only do so as a deliberate choice for special effect. Your normal practice should be to slightly mute the string as your finger moves.)
- At that point you are either in a new box or at least about halfway there. Here Tama's diagram is illustrative, as she shows various points in a scale at which you might slide on your way up or down the neck. (To go down the neck, you just reverse the direction of the slide.)
- If you are only halfway to the next box, do another slide to get all the way there. I find this clumsy to do on the same string I just finished sliding on, so I always do this on the next string with a different finger.
- Once you get to the new box, continue playing subsequent notes with your normal fingering for that box until you either finish the passage or until you need to slide again, in which case follow the slide procedure once more.
Ramillies gives you a different method--instead of sliding, which necessitates staying on the same string and moving your finger up or down it, Ramillies advocates using open notes to jump from position to position. However, s/he fails to break down what that actually entails (although, once again, the accompanying diagram does help after you understand what's being talked about). What you'll be doing is using the open note as a kind of trampoline to jump from one box to the next. To use that method, here's what you need to do:
- Register which open notes occur in the scale in which you're playing. (Or, more properly, which open notes occur in the solo passage--the notes in the passage might not include all notes in the scale, and conversely the solo might include notes outside the scale. For the sake of simplicity, however, these instructions assume that your passage does not include notes outside the scale.)
- Register which of those notes also happen to occur in the scale box from which you begin (your starting box).
- Given the position of those notes in the starting box, decide which open note makes the most sense to use as your "trampoline". (For example, if one of the open notes that happens to be present in your starting box also happens to come directly after your starting note, it probably doesn't make much sense to use that open note, as you'd only play a single note in your starting box before playing the open note and jumping away.)
- Play up to the note in the passage just before that open note.
- Now play the open note you decided to use as a "trampoline", and simultaneously jump your hand up (or down) the neck, placing it at the next box.
- Now play the next note in your passage with the correct finger for the new box.
- Continue playing subsequent notes in the new box, with your normal fingering for that box, until you either finish the passage or until you need to jump once more, in which case follow the jumping procedure again.
As Ramillies points out, his/her method has its advantages, but it requires more thought so is more difficult to apply on the fly when improvising; and sometimes (though admittedly rarely) the open note may be two strings away from its predecessor or successor rather than just one. In that case it's still OK for fingerstyle but if you're using a pick it requires a string skip, which IMO negates the advantage of avoiding the slide. As a result I've used it when playing classical or flamenco passages, but I've never bothered with rock/jazz (electric). Still, it's your poison to pick.