What's the purpose of stirring the mixture at 0 degree and dding NaOH to the organic reaction mixture?

I used to follow the lab protocol without thinking, but I now hope to understand why certain procedures are done.
I am synthesizing dimethoxyacetic acid from methyl dimethoxy acetate(so hydrolysis from ester to COOH). 


1. Add lithium hydroxide monohydrate (89.5 mmol, 3.75 g,1.2 equiv) to a solution of methyl dimethoxyacetate (74.6 mmol, 10 g, 1 equiv) in dioxane (45 mL) and water (45 mL) at 0 °C.
2. Stir the mixture for 1 h at 0 °C and one additional hour at room temperature.
3. Add sodium hydroxide (1M aq soln, 75 mL) to the mixture.
4. Extract the mixture with diethyl ether (375 mL).
5. Acidify the recovered aqueous layer with hydrogen chloride (6N aq soln) until a pH of 1-2.
6. Extract the organics with diethyl ether (375 mL).
7. Dry the combined organic layers with magnesium sulfate and filter.
8. Evaporate the solvent slowly under reduced pressure to obtain 2,2-dimethoxyacetic acid.


I thought the reason why you acidify the aq. layer was to convert -COONa or -COOLi (ester salts) in the layer to acid (-COOH). Then doesn't it mean that the product I want ( dimethoxyacetic acid ) will be in the aqueous layer? Why do I need to care about the organic layer?


Would appreciate any insight&help!


