Is it permissible for an employee to leave his work during official working hours, with the approval of his direct supervisor, for another job?

It is not permissible for an employee to engage in work other than that assigned to them during official working hours, except with explicit permission from the employer or their representative. Otherwise, this would be considered a betrayal of trust, based on the Almighty's saying: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence]."