I think you are still underwatering your kumquat. This is quite a big plant in warm conditions and will require more water than you are giving. It is quite possible that the centre of the plant is bone dry and that the water you are giving is only wetting the surface and the rest running down the edges of the pot.

Put the whole plant in a bucket and fill to the top of the soil level with a measured amount of very slightly warm water. Leave to soak for several hours. When you take it out, you can measure how much water is left and how much the pot has absorbed.

Many people are scared of citrus root rot from over watering, but under watering will kill much more quickly.

The other thing I notice is that some of the leaves on your plant appear to have white stippling. It could just be the photo, or it could be a sign of red spider mite damage. Inspect the leaves carefully to confirm and if so take appropriate action!

I have checked the leaves and they are perfectly clean. The white areas are probably just due to minerals in water.

I could be under watering it, so I am going to give it some more water considering its condition isn't improving. Thanks for the help.

It may be an underwatering issue, but it just may be a kumquat issue ;). The same thing you described happened to my big Nagami tree last year and it died. Now this year, two other Nagamis are doing the same thing and my Fukushu just died exactly like that....browning leaves, branch dieback and now it’s dead. This was a perfectly healthy plant two months ago. I don’t know what to tell you except you may want to try a calamondin instead ;). Good luck!

Thanks. I actually got a calamondin too when I bought the kumquat but it's much smaller. It has dropped a couple of leaves but it's nothing to worry about so far. It's doing much better than the kumquat.

I have watered the kumquat from the bottom as citrange2 suggested and I'll see how it goes from here.

There's a thing that citrus do sometimes, where if they are overly dry, they drop leaves just after they've been watered. If it was really dry when you bought it, hopefully that's all that's going on and it'll recover just fine.