One specific idea for using wider intervals in the beginnings and endings of phrases is to land on a chord tone and leap two chord tones up or down. This is effective at both the beginnings and endings of phrases or as a standalone melodic motive. All the great improvisers use this idea.
The example below shows what I mean. Over a C major chord, you could play a C and jump down to E (skipping the closest chord tone, G). There are three notes in the triad, so there are only six possibilities (R down to 3rd, 3rd down to 5th, 5th down to root, root up to 5th, 3rd up to root, 5th up to 3rd).
There are also many ways to embellish this idea, with approach tones to the first note or to the second note. Any of the typical embellishments can work on either note (diatonic or chromatic approach, enclosure/encirclement, neighbor tone, etc.). I show a couple of basic examples below.
A good exercise is to pick a particular leap and practice it with a particular approach or ornament through all the changes of a tune. Trying to land directly on "1" each time also helps practice forward momentum by hearing and directing your lines toward a conclusion. Here's example showing one variation, using a common chord progression:
Of course, you would want to create further variations, using the other possible leaps and varying both the approach and resolution.