DD has been wearing last year's coat which is far too small for her - I've struggled to find her something to replace it as she's small and seems to have very short arms so most coats drown her.... However I have hit the jackpot with a Gap padded jacket which looks like a perfect fit. Unfortunately though dd is 22 months old and seems to be learning to assert her independence with clothing choices - hence she absolutely refuses to wear the new coat and will only put on the old tiny one.
Put the old one away and give her a choice of new coat or no coat, she'll prob go with no coat so take the new one with you and if she's anything like my ds1 was at that age (and still is sometimes at 3.5) she will continue to refuse to wear it for a while during which time I would emphase how nice and warm it is with a coat on and then at some point she'll give in and want her coat on. It's not freezing so she won't come to any harm without a coat on for a little while and she'll learn why you need to wear a coat - that's what I would do anyway and worked well for ds1.
DS did a similar thing last winter with his new coat (he would have been around the same age). I just said if he didn't wear his coat we were not going out. He LOVES going out, so once he figured out that I meant what I said, he would tolerate me putting it on him. I sometimes have to wrestle him into it while he sobbed (if we had to get somewhere quickly).
I was anticipating similar issues this winter, luckily he has been fine
You need to hide the old coat out of sight, and explain to her firmly that the old coat is too small and has been put away because she will get cold. I would then take her with you to choose a new coat. Take back the GAP one and present her with choice of a couple of coats in the shop and let her choose. she may still whinge about the old one for a bit but will be less reluctant to put on the new one if she has chosen it.
DD1 (2.9) has decided she is very fashion conscious. She is also very stubborn! she loved her old winter coat and point blank refuse to wear the beautiful puffa I had bought her (also rom GAP!) I could not be bothered with a battle every time I left the house so took it back, we went on a shopping trip and she chose a lovely different coat which she is now excited about. I know 2.9 is different from a 22 month old, but I also have a 19 month old and she would be capable of understanding that old coat is gone and new coat which she likes is here to stay. Good luck!