You need to know the chord progression of the number you're playing. In jazz, you generally see ii-V7 or ii-V7-I. Another popular progression is I-vi-ii-V otherwise known as the doo-wop progression. When the Roman numeral is in lower case, that chord is played minor and when it is upper case, it is played major. V7 is the dominant 7th chord. It has a major root-third-fifth structure but with a minor 7th on top. This is the chord you will virtually always resolve back to I with. Its major-minor structure also allows you to change keys. Extremely important chord!
Blues is generally I-IV-V but this is a variation of ii-V-I (IV is sometimes substituted for ii and vice-versa). All these chord progressions come to us out of classical music where they are all found in abundance.
So, take a C major scale: C D E F G A B C'. C is I, D is ii and G is V7. You would do the same for any scale.
Now here is a typical jazz chart:
Notice that first bar is labeled as D-7 which is the D minor 7th chord. The next is G7. We want to find the key center of these two. Whenever you see a minor chord followed by a major chord, it's dead giveaway that it's ii-V7 or ii-V-I. So if D is ii in the scale, what is I? It would be C. And what's the fifth position on the C scale? That would be G and the fifth position is always a major scale with a minor 7th so it is G7. So in the first bar, the notes you have available to you are D, F, A and C for the D minor 7th part. The next chord, G7, gives you the notes G, B, D and F. You don't have to play all these notes, these are mainly what is available (there are others but we won't go into that now). But notice that both chords have D and F in them so you might want to concentrate on those two throughout the first two bars.
The third bar shows an E minor 7 and an A7. It's still ii-V but the key center has changed. If E is ii then I is D major 7 so the key center is D major. The fifth of the D major scale is A so we play A as the dominant 7th or A7. In the third bar you have the following notes available to you--E, G, B, D, A and C# but you might to emphasize using the E and G since they are common to both chords.
And you go through the whole number that way. Now you can just play the notes that are written on the staff which is the melody but that's really a reference--something to build off. You don't have to play it all if you don't want to. You don't want to play it straight because it's stripped down and pretty boring.
Now there are little tricks used in jazz--various note patterns and what not for handling certain situations. I don't know what they are for sax. I'm a double bassist. Yes, I use this same scheme to walk the bass. When you hear a bassist walking it and you wonder how he knows what notes to play and why it's never the same thing twice--he knows what chords he's playing at any given point and is choosing different notes each go-around so that it's always different. For sax, though, particularly bop, you're cramming so many notes in a space that I'm not at all sure how that works.
Now, if you're not playing off charts, you need to make some!!! Jazz is chart-crazy! Every professional jazz band plays from charts. There are even books crammed with charts for every jazz standard out there. These are called "fake books." The best fake book out there is called "The Real Book" (irony intended) and you can order it off Amazon. You have to have it! Your band MUST get their arrangements down through charts!! I cannot emphasize that enough. Once you get used to the charts, you can go out there and wing it without anything and sound great. But if you have no charts to work from first and you're just standing there trying to solo--it will never work. Charts help you visualize how a number is structured so that you know what notes you have available at any given point in the piece.
Hope this helped. If you have any questions, don't hesitate to get in contact with me. I'm here to help.