Perfect pitch is a great boon for a Jazz musician. Unfortunately it's uncommon. If you don't develop it at a young age, you probably never will. Relative pitch is being able to figure out pitches after you have a reference pitch given to you. I was able to teach myself what I call 'relatively perfect pitch'. This is something that is somewhere between perfect and relative pitch.
If I'm listening to notes played on my main axe, the alto saxophone, I can tell what pitches they are. I've been able to do this since I was in high school. You might call this something like 'perfect alto sax pitch'. I learned 'relatively perfect pitch' years later. The way I did this was by choosing a few pitches at a time (these might be from the melody of a favorite tune) and sing them over and over in a very loud voice. I'd sing these four to six notes very slowly and just notice what my body felt like as it vibrated these notes. If you sing loud enough you can learn to remember what your body feels like with these notes rather than just what they sound like. It becomes a physical sensation that is very hard to forget. Eventually you will ingrain several of these note combinations into your aural & body memory. You will then be able to relate any pitch you hear to these memorized tones.
I tell my students that each note is like someone you know. There is just one guy named A natural and he has a unique personality. You don't recognize him just by the facts that he has brown hair, green eyes, wears glasses, and is stocky. If this this was how you defined him you'd be calling all kinds of guys by his name. There is only one guy named A natural. Musicians don't know each note personally but they will get very close, without ever really locking the notes in. They may know A natural as somewhere between 438 bps and 444bps, but they have never really sat down with 440 bps and gotten to know it personally. Once you sit down with someone and get to know him (find out where he grew up, what his parents were like, what motivates him) and find out what you have in common with him (how your head, throat, chest feels when you sing the note for periods of time), then you will probably never forget him when you pass him on the street. You won't say," That dude looks really familiar, is it because he looks kind of like my other friend G#?". You will know this old friend immediately and say," A440! Whasup! Haven't seen you since the bridge."
This is not some magical process. It is just a matter of taking time to sit down with these notes and spending some quality time with them. After you work on your note combinations during your practice time, remember to sing them occasionally during the rest of the day. This will help solidify your personal knowledge of these notes. First try to sing them in your mind. Then check them by singing them in your full chest voice. If you're patient and diligent you will be guaranteed results.
Thank you to Andreas Steffen in Germany for suggesting this topic. Sometimes after dumping my brain night after night I am at a total loss for my next topic. Any topic relating to Jazz, saxophone, the music business, harmony, or esoteric music that you would like to read about please let me know.