Soap is a fatty acid salt of an alkali metal, typically sodium or potassium. The fatty acid itself is always insoluble in water, and therefore acidification of the soap solution produces two layers, a lower aqueous layer with the alkali salt of the acid and an upper layer of fatty acid laced with aqueous material. This can be treated with an organic solvent to dissolve the fatty acid and expel the aqueous material, giving two immiscible layers.
I suspect that your soap is a combination of fatty acid soap plus SLES.
Sodium laureth sulfate is not a soap, it is a surfactant, and it is so constructed so as to be more soluble in water than sodium laurate (which is a soap) or sodium lauryl sulfonate (which is a surfactant) by addition of typically 3 -OCH$_2$CH$_2$O- (ethylene glycol) groups. This complicates the precipitation and extraction; another variable is the solvent: while petroleum ether is organic, it has poor solvent power, and long chain acids are not well dissolved. That's why you have to do the extraction many times.
I visualize your experiment to go this way:
When you add the concentrated HCl to the sodium laureth sulfate, you insolubilize the HLES (the acid form), but it is not easily separable. A variable here is whether there is excess acid and how much. Another variable is that the precipitation is incomplete.
You add petroleum ether to extract the HLES and reject water and excess acid, leaving, hopefully, mostly NaCl in the aqueous phase. You do this again, extracting more HLES.
However, HLES is somewhat soluble in water (because it contains 3 -OCH$_2$CH$_2$O- groups), and is quite a good emulsifier for organics, and pet ether has a lot of low mw molecules, so before the last of the HLES is extracted, you find that the HLES has attracted a lot of water and a lot of pet ether. But this combination cannot contain all the water, so it is immiscible with the lower aqueous layer; it contains too much water to be miscible with the pet ether, so it is a third phase. Excess acid in step 1 complicates things because acids increase the solubility of ethoxylated surfactants.
You get something like this:
So instead of dealing with a soluble/insoluble situation, you have a significant but incomplete precipitation of HLES, and what doesn't precipitate out manages to emulsify with some of the pet ether and create the third phase.
A possible way to improve your separation is to add NaCl to the mixture after adding the HCl to maximize the salting-out effect. Another adjustment is to minimize excess acid (add just enough acid to precipitate the HLES - measure pH with paper) or add NaOH to adjust pH to near neutral (pH 6-8), when you get the third phase. This minimizes acid solubilization of the HLES, but keep the aqueous layer saturated with NaCl by adding solid NaCl if necessary.
You could also substitute toluene for the pet ether, but there is somewhat more hazard. Diethyl ether would be the extractant of choice many years ago, but alas, we must emphasize safety.