The easiest, or minimum, way to make a transition a half step up (with transpose button) would be at least to insert the dominant seventh in the new key before starting the song in the new key.
E.g. Final chord of song 'C', then 'G#7', then 'C#'. So, press transpose button in the moment you start playing 'G7' so it gets 'G#7'. That would be a typical elementary shift upwards in thousands of pop songs.