In soil, oxygenation isnt an issue the same way it is in hydro.
Its good to leave the water sit to get rid of the chlorine, but oxygen levels will not go down to any degree you need to worry about.
When the water gets into the soil, assuming you are not over watering, it gets spread out over the surface of countless small particles in very thin layers. Thin layers of water will reach 100% dissolved oxygen (DO) very very quickly - even if they start out at a low DO. Thicker layers of water are slooooow to reach 100% DO.
BUT - the key is to have soil with lots of air spaces in it. Thats why its recommended that you add 20% -40% perlite or vermiculite to most soil mixes - so they can breath better.
Both of those increase the air spaces in soil and allow for much better aeration. That and dont over water. Too much water forces the air out of the soil - which means less aeration - which is bad.
This is also why fabric pots or air pots are good - better oxygenation of the soil.