Dressing baby for pram walks in autumn?

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Silly FTM question but what should I be dressing my 2 month old for pram walks now it’s getting a bit colder? When it’s been warm I have just been taking her out in her indoor clothes with a blanket handy if needed. It’s still 20 degrees here but felt a bit chilly today and I realised she might need more clothes for going out. At what point does she need a hat and do I go straight to a woollen one or are the cotton ones that come with some of the baby outfits fine for Autumn?

We lived in Sweden and everyone there has a sleeping bag type of thing for the pram, and baby has normal indoor clothes on, and is all warmed up in the sleeping bag. They have very thick one for freezing winter and lighter ones for milder season.
Wool hat is better. A light quality one is good for autumn. Some have a been cotton/wool

Where are you? I'm in the Sae and had beautiful oram suits, never needed!

Thanks for the responses! Probably time to buy a wooden hat as I only have the cotton
ones so far. Tried one of those yesterday when it seemed chilly and she just wiggles out of it.

I’m in Cambridge so it’s not too cold yet. I was wondering about pram suits as they seem awkward to take on and off if you go inside a cafe or something.

Are the sleeping bags used in Sweden similar to night time sleeping bags sold here or specifically designed for the pram?

You can get pram suits for when it's much colder - like a sleeping bag but with arms and legs. Also, tuppence and crumble star suits are great.

My baby was a November baby and it snowed when she was a week old, we just layered her up and kept checking she wasn't too cold or too hot. Remember if you use blankets as well they can easily be removed.

You can get specific pram sleeping bags, I think they’re also called foot muffs.

We had a Tuppence and Crumble starsnug wrap for ours last winter and will probably buy the next size again this winter. It’s probably not as warm as a sleeping bag but we found it great as we could also use it with a carrier and in the car seat.