How would one describe tightly pulling on your coat to feel warmer in winters?

The word I would use is snug, either as a verb or an adjective:

[Merriam-Webster]
intransitive verb
: SNUGGLE, NESTLE
// snug down for bed

transitive verb
1 : to cause to fit closely
2 : to make snug
4 : to secure by fastening or lashing down

adjective
1 c : fitting closely and comfortably
       // a snug coat
2 a : enjoying or affording warm secure shelter or cover and opportunity for ease and contentment
       // a snug cottage
       // a snug haven

So:

  • I snugged my coat around me.
  • My coat was snug and warm.