To my ear the most usable note outside the major scale to use as a approach tone, chromatic movement or filler note is the b3rd. I use this interval all the time to help break the monotony of standard major scale lead lines.
Below are six scale diagrams. The first one shows the 1st position of the major scale. In this example the white dot represents the root note of the scale and the red dots represent all the other notes of the scale. The next diagram is the same scale only this time the b3rd note is represented as a yellow dot. The four other diagrams show the same type of thing for the 3rd fingering of the major scale and the 6th fingering of the major scale.
The TAB below shows two small examples of how I often use the flat 3rd in a major scale melody line. In example #1 I use the b3rd as a approach tone leading to the 3rd and directly followed by the root note. I love this cadence, use it all the time it really opens a jazz feel to your lead line. The #2 example is very similar but the cadence uses the b3rd a little differently. This line should sound familiar to you as it is used in a lot in rock, jazz, blues and bee bop situations.