Part of the rice trick is to put dry rice in the ukulele, tape the sound hole with blue painters tape, and then shake it all around to clean off the inside of the ukulele a bit. You will be amazed at how much dirt and filth this will remove from the inside of a dirty uke. I have done it with three ukuleles and it made a world of difference.
Also, have you really cleaned the outside of the ukulele really, really well? I had a uke that arrived just reeking of cigarette smoke. I cleaned the outside of the uke several times before I got all the stench out, but then when I played a song, stink just came shooting out of the sound hole. That is when I went to the rice trick. It worked extremely well, and I had to do it three times before the rice came out clean.