I recently wrote some pointers about improvising on the forum, in this post, and thought to repeat it on the blog. Hope you find this useful!
Jazz players usually improvise not just by thinking about melodies, but also by choosing notes that go beyond the basic 1-3-5 notes in a chord. For instance, instead of using the usual C-E-G notes during a C chord, you can play a B and make the chord a Cmaj7, or B and D to get a Cmaj9 chord. This makes the solo sound more unpredictable and sophisticated, and of course melodically more interesting. At the same time, by focusing on chords instead of melodies you don't stand out and distract the congregation during for instance free worship.
Here's an example, using the the first 16 seconds of the Yahweh improvisation I did some time ago; the video is embedded below. It sounds like I did a lot of planning and thinking beforehand but this occurred on the fly as I translated what I felt into music - just like revolving your melody to the 1-3-5 notes of the chord can be natural, with practice and exposure you can do the same for more complex chord patterns.
The chords that are used in the first 16 seconds are E, F#, C#m, G#m, E, F#. I list the chord equivalents of the improvisation, and the exact notes are put in brackets. The intuition is also below.
F#6 (D# F#)
C#m11 (B F#)
Here, I did not alter the E chord. The idea is to start with something simple, something like a "launch-pad", to introduce the improvisation. It also gives me a little time to get a 'feel' of the song. I then moved on to more complicated chords - F#6 instead of F# for example - to create a feeling of dissonance. This makes the melody move forward as the listener waits in anticipation for the melody to resolve and better harmonise.
G#m7 (D# F# [there's E too but its just a passing note between D# and F#])
Emaj7 (D# B)
Some resolution is provided here, but not too much. I used 7th chords, which are quite close to plain vanilla chords. 7th chords are also quite soothing, especially when they are used on minor chords and IV chords (key here is B, so the IVth chord is E).
This is the end of the section, but there is a long way more to improvise. Hence I wanted to drive the music forward a little more, and used a sus chord. The note G# does belong in F#, but it is very close to F# and A# which are strong defining notes of the chord. To the listener, there is a feeling that G# will move to either note soon. However, this doesn't occur, leaving the listening hanging, and you kinda keep your audience interested awhile longer.