Is it permissible for my husband to use the company car belonging to Saudi Telecom Company (STC) for extra work, by his own effort, to increase his income, knowing that he covers the fuel expenses from his own money?

If the laws and regulations do not prohibit the husband from working another job in addition to his current one, and the company allows him to use its car for his other work, then there is no harm in that. However, if the regulations do not permit it, then he is not allowed to violate them. Furthermore, using the company car is not permissible without the company's permission, whether he pays for the fuel or not.