From a top bunk in a 12-person prison cell in Fort Dix, N.J., Martin Shkreli is at work on a big second act.

Wielding little more than a contraband smartphone, the disgraced pharmaceutical executive remains the shadow power at Phoenixus AG, the drug company that became a national lightning rod for jacking up the prices of rare drugs under its former name, Turing Pharmaceuticals AG. Mr. Shkreli still helps call the shots. A few weeks ago he rang up his handpicked chief executive during a safari vacation—to fire him, according...