What Susie said. Some other thoughts --
Vessel size -- If I'm calculating things right, you are making roughly 2 gallons (8 liters) of soap paste in a 20 gallon (80 liter) container. If I'm correct on this point, you are using wayyy too big of a vessel for that amount of soap paste -- you need to be making this in more like a 5 gallon bucket, not a 20 gallon vessel. For efficient conservation of heat and the most efficient mixing, you want your soap batter to be as close to a sphere in shape -- not spread out like a rain puddle on the sidewalk.
Intensity of mixing -- you will get more efficient and complete mixing of a liquid if the liquid is deeper rather than shallower. Furthermore, you've dropped your intensity of mixing by switching from a stick blender to a paint stirrer. That's maybe something you'll have to live with, but you're going to have to understand that a paint mixer is not going to work well when you eventually scale up to a full 20 gallon batch. Most people using that size of vessel are using an industrial "stick blender" not a paint stirrer. Maybe something vaguely like this:
http://www.zesco.com/Dynamic-SMX800E-Commercial-Immersion-Blender-27-Shaft-pz523D005.htm
Think about it -- If your smaller-scale soap making equipment is anything like mine, the batter in your small soap pot is about as deep as it is wide. That's what you want for your scale up conditions too. You're going to keep the batter at a more consistent temp and you are going to spend less energy and time getting it emulsified if you do this.
If you really want to get the best out of this investment, then I suggest you consider getting away from the paste method of making liquid soap and go to an "all in one" method where most or all of the dilution water is added up front. This method has little or no dilution at the end -- your batch size is fairly consistent from start to finish. Further, you won't have to deal with the thick paste which can be difficult in small batches and a terror as a large batch. With this type of process, you can use a lower intensity continuous-duty mixer and just plan on mixing for a longer time. Although some people do this kind of "all in one" liquid soap making on a hobby scale, it's more practical on a larger scale.