Saudi Arabia is quietly trying to mend fences with Iran and other regional foes as officials in the kingdom have grown more worried about the risks conflict poses to its oil-dependent economy.

Riyadh’s newfound interest in better relations with regional rivals comes as Saudi officials question how much backing it has from the U.S. and other allies. Saudi calculations changed after a cruise-missile and drone strike—blamed on Tehran—temporarily disabled a large portion of the country’s crude production earlier this year. Washington...