This idea is speculative. As far as I know, there is as yet no experimental evidence to support it. For example, it is not known for any particular reaction mixture how much of that mixture could be mixed with how much of the hexane-methanol azeotrope and still actually get the essential separation into the two phases. Also, even if liquid layer separation is achieved, we cannot know in advance what the difference in partition coefficients between mixture constituents will be even though this will determine how practical the separation will be and how many times the procedure would need to be repeated to get a useful concentration of even one species.
The azeotropic mixture of hexane and methanol splits into two phases when cooled below 35 C. At temperatures below the UCST there exist two phases containing different proportions of hexane and methanol. The azeotrope between hexane and methanol has bp 50 C; its composition is 73.1% hexane and 23.9% methanol. The relative volumes of the upper and lower phases are 67.8: 32.2, about two parts to one part. When the hexane-methanol separates on cooling the composition of the upper layer will be 85% hexane and 15% methanol with a specific gravity of 0.675. The composition of the lower layer will be 42% hexane and 58% methanol with a specific gravity of 0.724.
A mixture of substrates (such as the untreated components of a reaction mixture) as evaporated to an oil or oily solid could be mixed with the warm single phase above 35 C, then cooled to split into two phases into which the mixture of substrates would partition. Subsequently, cutting the phases and evaporating each would give two new compositions with a different ratio of the substrates from each of the two hexane-methanol mixtures.
These substrate concentrates could then each separately again be dissolved in a new portion of the single phase azeotropic above the 35 C UCST and again cooled and the phases separated and evaporated. These oily residues are treated just the way solutions are treated in a liquid-liquid extraction to improve the degree of separation of the substrates.
The disadvantage of the methodology will be that each time the extracts need to be evaporated essentially to dryness because otherwise the proportions of liquids hexane and methanol will change so that the two phases will not continue to separate.
The advantage is that because both phases contain the same two solvents-just in different proportions, the partitioning of substrates between the phases might be expected to be closer to 1:1 and so the selectivity in the partitioning of two similar substrates might be more sensitive. That is to say, one substrate might be slightly more soluble in the methanol-rich phase while the other might be more soluble in the hexane-rich one.
If no separation of liquid phases occurs at any stage it is possible that a larger proportion of the hot azeotropic hexane/methanol mixture is required. This is easily rectified. Simply add more of the hot azeotrope composition and rewarm the total solution until above the UCST and recool.

