Instead of a part time job, call yourself a "consultant" and say you have "1/2 of your time available". Companies are generally more likely to bring in a consultant/freelancer 1/2 time than an employee.
(Make sure you charge enough though - being a consultant means you'll have costs you're not used to as an employee, like insurance and self employment tax).
(Make sure you charge enough though - being a consultant means you'll have costs you're not used to as an employee, like insurance and self employment tax).