your have the aloe in a fairly big pot in what looks like potting soil, which would be o.k. if you do not over water.
Ideally, you would select the pot size based on the size of the root and not on the size of the top. The pot should just fit and breathable pots are better than plastic or glazed ceramic. Think concrete and unglazed terra cotta.
Cactus mix is better than potting mix, but if you tend to over water and you have a top heavy plant, go with cinder or pea gravel. Even no. 3 blue rock will work. The pot will be heavy to move, but the plant won't topple over and it drains well. Pea gravel and blue rock can usually be gotten at a building supply either by the bag or the truckload. Because it will not retain nutrients, use osmocote every three to four months about a tablespoon is enough.
When you water, do not water with the saucer under the plant, either take the pot outside, or put it in a tub to drain. If you get a plant dolly it will make it easier to push around.
I think you should repot the plant again in a smaller pot.
Take the plant out and wash off the soil.
Check the root for signs of rot, mushy, smelly, soft tissue. If it is only the roots and the crown is intact, you can cut off the dead roots. If the main stem is rotting, more drastic measure need to be taken.
You would have to take off the lower leaves and find a part of the stem that is still firm and healthy and with a clean sharp knife, cut it through.
Now, this is important cacti and succulents need to callus off before replanting. Discard the diseased roots. If the leaves you cut off are still good, you can keep them, just make sure they have been cut cleanly.
Let the good leaves and the part of the stem you cut off air dry on a few sheets of newspaper until the cut ends have callused off. No water is required. How long it takes depends on conditions but a few days to a a week is usually good.
In the meantime, look for a pot that is deep enough but only a couple of inches wider than the root mass or stem.
Clean out the old pot with 10% bleach. You can dry out the soil and use it for something else.
Get the rocks, cinder or cactus mix.
When the ends of the cactus and the leaves have callused, then pot up the plant stem in the new pot make sure it is deep enough so that it is held firmly. Stake the top if you have to, but in most cases it does not need it. Take the leaves and put 2-3 inches of cactus mix in the old pot and stick the leaves in the cactus mix. Water everything until the water comes out the drain holes and then let them driain in the tub or sink for at least 30 minutes before placing them about 18 inches from the light source. Water them again only when the soil or rocks are dry a couple of inches into the pot.
The leaves will take a while and they may look shriveled, but a new plant will grow from them. The aloe will also do the same thing and the leaves will look a bit shriveled. The plant will be sustaining itself on the remaining leaves until it can make new roots.