As you noted, much is used with the comparative form of adjectives. You can understand and remember this usage by thinking of it as by much, i.e. as by a significant margin. In other words, much colder means colder by a great deal.
This usage also applies to the superlative of adjectives, and it is done by adding the, in order to make it singular. Examples:
much the easiest means by far the easiest
much the strongest
much the fastest etc.
Much is also used with what I think of as past participles of verbs, used as adjectives. Apparently a lot of people now call these passive forms of verbs, but I prefer to think of them as past participles. This probably just means that I am old or a simpleton or both. Examples:
the much abused word ...
Picasso's much admired paintings ...
much given ...
That last one is an idiom and has an illustrious pedigree. For example, King Edward VII was much given to gluttony.
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