According to his post office employment/payroll records he was promoted on April 1st, 1954, which would be just before he went in to the hospital. So when he came out of the hospital he certainly could have done some other job(s), but if that was the case, why did the post office then promote him again in March of 1955?
What's odd about the timeline in 1954/55 is that everything seems to point to the hospital incident being in April 1955 - resigning from the post office in "ill health," working the non-post office jobs in the second half of 1955, etc. But we know from the letters and bills that the hospital stay was in 1954. So that mid-50s period has a few question marks.
What makes the argument for the uninterrupted period at the post office is the promotions. Think about it; if you worked a job for a couple of years, then left just for the hell of it, then came back, would you be promoted after you came back? Maybe that's how the post office worked, I don't know. But if you left for medical reasons, I can see them "holding" your position for you until you came back.
What could have happened is after the hospital Bukowski wanted to leave the post office, tried some other things (like driving a truck), and then figured that going back to the post office was easier since he was getting established there. But once he got back in he was disillusioned again, and quit (right after receiving a huge raise, interestingly).
Speculation, I know, but it's all we've got unless there are more letters from that period waiting to surface.