M.B.
I see that you have already received some good advice. I also noticed that you are in Richmond. I am a teacher in your area, and I know many other teachers in the area as well. Several of them tutor for free or for a very reasonable rate. If you are interested, you are welcome to e-mail me privately. I know teachers at many of the Lamar CISD schools and might be able to help you find a convenient (and affordable) tutor.
Also, have you gotten any information about the extended day program at your son's school? All of the elementary schools in the area, to my knowledge, offer an extended day program for those students not performing well in reading and math, sometimes other subjects as well. Some schools are already holding extended day, and other schools are waiting until closer to the TAKS test.
You may not be aware of it, but the district's third graders should all be taking a practice reading test some time this month. For students who do not do well on the practice test, the district will probably be providing tutors called ARI tutors; they have done so for the past several years. The district ARI tutors generally start in January. Your son's teacher will probably get results of the practice test in December or January. He/She can then tell you your son's score and also which objectives are giving him the most trouble.
The bad news, as you stated in your post, is that he is required to pass the TAKS Reading test for promotion to fourth grade. The good news is that the state will offer three chances to do so. If your son doesn't do well in February, he will get another opportunity to retake the test in April. The third opportunity is during the summer.
I hope this helps!