It was a stunning moment for Exxon Mobil Corp. and the wider corporate world: A tiny activist fund had succeeded in changing the company's board.

But in the hours leading up to this week's annual shareholders meeting, Exxon went to extraordinary lengths to head off the threat from a campaign about which it had been largely dismissive months earlier.

Exxon telephoned investors the morning of the ballot—and even during an unscheduled, hour-long pause during the virtual meeting—asking them to reconsider their votes, according to several of those who received calls. Some said they found the last-ditch outreach and halt to the meeting unorthodox and troubling.