I also don't think it's fair for a part-time employer to expect you not to supplement your income. If you've explored full-time work with them and it's not available, then this is your option. Talk to your supervisor. Get his/her buy-in and support. It will ultimately make you look better to both your current employer and your prospective employer - or it should.

If they worked in retail, for example, they were told that it was unacceptable to have another retail job or if they worked in a restaurant that it was unacceptable to have another restaurant job because their loyalty was in question.
If either job has hours that are flexible at the employers discretion, you have to make sure that both places will work with you to work around your schedule. Even if a business does not expressly say that they are unhappy that you are working another job, they can use the schedule to make it difficult or impossible for you to keep both jobs.

One co-worker is very open about her other job and makes sure the scheduling supervisor knows when she is available. Another employee recently quit in the middle of his shift because according to him, he had repeatedly been scheduled to work when he had informed the supervisor that he was not available.

One possible difference is the first employee mentioned is an exceptional worker; the second one, not so much. I might add, though, that he is working three jobs and going to graduate school, which certainly could play a role in both his lackluster performance and difficulty in scheduling.

My supervisor knows that my husband is an unemployed alcoholic and that my pay is responsible for paying the bills, so perhaps transparency is best in this case.