It may be the amount of sun that portion of your lawn is receiving. It may also do with how deep the topsoil is underneath. St. Augustine is also prone to brown patch or take all diseases and adding fertilizer often makes both of these worse. Texas A&M has found that adding a thin layer of compost helps combat the disease problems. The link below is to a TAMU article on St. Augustine written in 2002 that might be helpful.

Here is a link that might be useful: St. Augustine Article

If it isn't a sun issue I would agree with denise that it might be the depth of the soil. Adding compost not only would help with diseases if you have them, but it will also help with the depth of your soil. You would want to add compost and top soil in thin layers over several seasons, working best when the grass is actively growing. Be careful to make your layers thin enough that they don't smother the runners. I've also noticed when I compost my lawn, I don't have to water as much either. The compost also feeds the microscopic critters and earthworms in the soil for better tilth - kinda like natural aeration. I'm not a completely organic gardener, but I do use an organic fertilizer twice a year for soil health. Unless you see that your grass has a disease, I'd work on the soil and that should improve your grass.

I have to hand water the less lush places in my yard, especially under dense parts of trees. Those parts do not get as much rain. Also, many of my neighbors do not have grass under their trees because they do not keep the leaves up and/or the lower limbs trimmed away. Your SA needs light and air....and water. I also use airiator sandles on my problem areas (great exercise!)...as well as regular machine airiating and composting. If you yard is older, some gypsum applied to your problem areas will help to break up the compacted soil and give it some TLC too.