Is it permissible to pay a bribe to the new work supervisor to temporarily overlook my working in two jobs, knowing that this is my sole source of income, and fearing dismissal from both jobs if the matter is discovered?

Before answering, there are several matters that need to be addressed: a woman's travel without a mahram (unmarriageable kin) is forbidden if there is a fear of fitnah (temptation/strife). An employee is obligated to perform their work according to the contract. Bribery is forbidden. If there is an urgent, compelling Shar'i (Islamic legal) necessity that leads to unbearable hardship, then there is no blame in committing what is forbidden, because "necessities permit the forbidden and are estimated according to their measure."