Prince had only a short gap between
Purple Rain and
Around The World And A Day, but there was a gap.
The gap was five weeks (4/13/85 through 5/11/85). The last single from
Purple Rain, "Take Me With U", spent its last week in the Top 40 on 4/6/85; the first single from
Around The World In A Day, "Raspberry Beret", debuted on 5/18/85.
Another near miss: there was a gap of only three weeks (10/8/83 through 10/22/83) between the last single from Culture Club's
Kissing To Be Clever, "I'll Tumble 4 Ya" (last in the Top 40 on 10/1/83), and the first single from their next album
Colour By Numbers, "Church Of The Poison Mind" (debuted in the Top 40 on 10/29/83). Note that although there was a gap, they still had singles from two different albums in the Top 40 during the same calendar month.
Rick Springfield had a gap of only one week (3/6/82) between the last single from
Working Class Dog, "Love Is Alright Tonight" (last in the Top 40 on 2/27/82), and the first single from
Success Hasn't Spoiled Me Yet, "Don't Talk To Strangers" (debuted in the Top 40 on 3/13/82).
This one probably merits an asterisk, but Duran Duran was only out of the Top 40 for one week (6/11/83) between the last single from
Rio (the title track, last in the Top 40 on 6/4/83) and "Is There Something I Should Know" (debuted in the Top 40 on 6/18/83), a newly recorded single which was included on a U.S. re-release of the band's self-titled 1981 debut album. In the U.K., "Is There Something I Should Know" was a standalone single not on any album, but in the U.S. the relative importance of singles versus albums had already swung far enough in the direction of albums that it had become rare for a major label pop/rock single to not be available on an album at the time of its release, or at least soon after. At the same time, while the band's self-titled debut album had attracted little attention in the U.S. when it was first released in 1981, Capitol Records saw the success of
Rio as an opportunity to promote it to U.S. audiences. Including "Is There Something I Should Know" on the re-released debut album killed two birds with one stone, making that song available on an album while providing a single that could be used to promote the re-released debut album.
Many of the artists mentioned so far fit the following pattern: 1) the earlier of the two albums was either their debut album, or their first album to really be a major hit, and that album wasn't an out-of-the-box hit, but took a little time to develop into one; and 2) all of this happened at a time when pop/rock artists were still expected to put out a new album pretty much every year, so they were likely working on their follow-up album before it had become apparent how long of a shelf life the earlier album would have, and the old album was still riding high on the chart when the new album was completed and ready for release. Billy Joel, Rick Springfield, Men At Work, Culture Club, and Madonna fit this pattern well, Andy Gibb and Duran Duran somewhat less so, Prince not really at all.
In Madonna's case, there was actually a last-second decision to delay her second album and release one more single from of her debut (evidenced by the fact that "Like A Virgin" has a sequentially earlier catalog number than "Lucky Star"). In Men At Work's case, the record label considered doing that but ultimately opted not to ("Be Good Johnny" was originally slated to be released as the third single from
Business As Usual, actually having a catalog number assigned before it was cancelled). That Men At Work was an artist from outside the U.S. whose first album had become a hit elsewhere sooner than in the U.S. was likely a complication. If "Be Good Johnny" had been released in the U.S., they would have been forced to either make the rest of the world wait for the new album (to the possible detriment of their commercial position outside the U.S.), or to release the new album outside the U.S. and somehow try to prevent material from it from leaking into the U.S. and competing with "Be Good Johnny". Neither option really made sense, so they moved on to
Cargo worldwide even though it meant cutting the legs out from under
Business As Usual in the U.S., where it was still high on the charts and looked like it could have produced at least one more hit single.