i have the case similar to yours, my maid gave me the resign letter on 15 Mar, she is a really good helper but she has the family problem need to quit the job, so i try to find the local maid, but they are so choosy, they want private bedroom or not willing to take care of newborn. Luckily, i found the new one who is pre-terminated by the employer due to financial problem(i have called her boss about her performance in this 3.5 year and check whether the financial problem is real or not, because the maid need the supporting document from employer), now all the documents & procedures almost completed. she can work with us on 24/4.
For special case, for the one who employer (1. financial problem 2. die 3. migration to other counytry) the immigration will approve them to stay in HK for processing the document and the time for procedure may be 5-6 weeks which is shorter than the oversea hire.
Or you may consider the local maid with finished contract, but they are too picky.