What is the general customer service number? the General Insurance Customer Support Service Phone Number. The customer support phone number of the General Insurance is 1-866-519-7422 (Click phone number to call).What is the telephone number for customer service? 1-800-566-0065, Customers can also use this number to reach to the customer service representatives of the company. 1-800-238-0271 is the fax number of the company, which can be used to send the orders.Is there a customer service telephone number for? General Customer Service: 1-877-383-4802. For faster customer service, please use the links above to contact the appropriate customer service center for your specific needs.