As I’m not overly familiar with the workings of the EIC I cannot really answer your question directly. However, I can provide you with some context on the issue.
It’s important to note that although European armies of the 18th and 19th centuries were formally dressed in heavy and elaborate uniforms, which might or might not include items such as long woollen coats, elaborate headgear, sashes, swathes etc., these uniforms were not in everyday use – certainly not to the extent where a private soldier would put on his full outfit when he rose, and take it off when he went to bed. These uniforms were often uncomfortable to wear and – much more importantly – very expensive. Whether these uniforms were bought by a state, a company or an individual, strenuous efforts were made to reduce wear and tear. To this end, most armies of the period would maintain one or more forms of working-, barrack, or garrison uniform. These would be made of much lighter (and much less expensive) materials (such as cotton or linen), and would often as not be grey or uncoloured. These uniforms would be used for everyday drill, or camp work, as well as when off duty.
You should also note that the concerns of private soldiers and NCOs were accorded much less weight than they would today. This is partly explained by the highly hierarchical way in which both army and society as a whole was organised, but also by the much lower degree to which individualism was valued and appreciated. Armies were seen as symbols of state power and prestige, and their appearance should reflect this. The uniform was not just designed to be practical – it was just as much a costume – mean to convey the prestige, status and strength of the owner. Thus, if a red, woollen coat was seen as integral to what a British or EIC soldier was, then those coats would be worn by the soldiers.
There are notable parallels to modern day society and armies here. Take formal dress – even the civilian versions. Many forms of these are not particularly practical or comfortable. Think high heels, tight dresses, white tie and tails etc. These are worn because of the image the wearer wishes to convey, because the wearer wishes to signify membership of a select group, or because “the occasion demands it”. Often all three at the same time. Modern day armies likewise sometimes have a tendency to acquire equipment or units as much for symbolic reasons, as practical ones. Think huge billion-dollar projects which serve as much as symbols of national prestige and power, as they do practical defence.
So although the heavy, madder red wool coats that we have come to associate with British and EIC armies of the early modern and colonial period were indeed worn extensively, it is a mistake to assume that these were the only type of dress available to a soldier, and that all his daily tasks would have been performed wearing these. Similarly, it is a mistake to conceive that these coats were originally mandated purely to meet standards of comfort and practicality. They were not, and even when such matters became pressing, change would rarely be instant.