The day after Sony Pictures employees discovered that company email was unusable following a cyberattack, senior executives came up with an old-style communication network: a phone tree, in which updates on the hack were relayed from person to person.

With computers and landline phones down during Thanksgiving week, the Sony Corp. studio’s 6,000 employees were forced to improvise, with cellphones, Gmail accounts and notepads. The payroll department dug up an old machine to cut paychecks manually. Before long, the studio unearthed...