I have an old Axiom with a removable hood that is pretty good. Never did use the hood much unless I was hoofin' it.
https://i.ebayimg.com/00/s/NzY4WDEwMj...WFg~~48_35.JPG
The MEC rain jacket I bought to replace the Axiom has no hood and I'm very happy with that but the still looks, feels and fits like cycling gear which led me to purchase a SWRVE Milwaukee coat for those wet days that are less monsoon-ish.
The SWRVE Milwaukee jacket also has a hood that I don't use while cycling but comes into play while walking about.
https://swrve.myshopify.com/products/milwaukee-jacket
I should mention that both hoods are designed to provide good visual peripheral clearance.

I just picked up a new jacket that has a hood on it. It's pretty useless and mainly flaps behind me, but it's a running windbreaker and the price was right.

This:
https://www.llbean.com/llb/shop/63187...-with-gore-tex
The hood goes under the helmet so there is no loss of peripheral vision. It doubles as a winter top coat. I've been using mine for ten years.

Shower's Pass jacket is what I use. One of the few things that annoys me about the jacket is it didn't come with a hood. Cost me an extra $25 for a velcro attachable one that is a PIA both for fit and the time it takes to Velcro it to the jacket. I've always used hoods on my rain jacket and never had peripheral vision issues. I wear them under the helmet. But I prefer the type that roll neatly into the collar when not needed and roll out easily for times you need it.

I like a jacket that I can use on tours or walk around with when not on the bike. The problem with a helmet cover is that it's fine while you're on the bike but I'm not too keen on the fashion statement it makes to walk around with an upside down bowl on my head while waking about and the bike is far out of sight.

Cure for flappy hood syndrome - pull the cords tight so it gathers itself behind your head

I have a hood for colder weather rain and wear it under helmet. In summer a tiny "Pocket Rocket" without hood.
For touring the hood is mainly useful at camp.

+1 for hood UNDER helmet.

Can't ever get any hood to not block peripheral under helmet :/

M.

Mine has one, I can roll it up, & tuck it away when I don't need it.
When its really wet, I'll use it. Its basically a tech jacket so its made to have helmets underneath. There's pull ties around the head, & another set around the neck.
This one doesn't block my vision, or catch the wind.
I could even wear it under my helmet if I really wanted to.

Rab Latok technical shell, made out of eVent fabric.
It does a great job of blocking the wind, while being waterproof, & being more breathable than goretex.

I wear a Layer 1 (base) with a neck zip.

I wear a Layer 2 (middle) with a hood and a neck zip:

I wear a Layer 3 (shell) the prevents air passage with hood and a neck zip as well.

This is Layer 2 and Layer 3 fits over with a hood and is more than enough for me down to 0F. Also my helmet fits under both hoods.

Yep! +1
Its all about proper layering.
I wear merino wool base layer, a thermal jersey,& my Rab.
This work well for 0°- -5°, anything colder & add another layer or 2.

My layer 2-3 have neck zips for moderating heat on the go.

I agree with layering. Each layer is quite inexpensive and gets daily use (€20 for Layer 1, €60 for Layer 2, €30 for Layer 3) depending on conditions.

I have a mild wool allergy where it slightly itches when it touches my skin. I can wear sweaters and jackets made of wool, but I'd hesitate at wearing a wool base layer. Although, it would be interesting to try it once or twice to see how it goes.

I do find the synthetic stuff through CRAFT quite high-quality and low-cost compared to most other brands which spend a huge amount on marketing ... I'm talking about you Under Armour

https://www.craft.se/products.html

I have a Showers Pass Elite 2.1 with the optional hood, but I leave it detached most of the time. The hood would make my helmet fit funny and block peripheral vision while cycling. However, I wear my SP jacket for street use as well, and often use the hood when walking in rainy weather. I keep the hood stowed in the rear pocket so it is handy to use when needed. I am really impressed with the SP Elite so far, and it is living up to the rave reviews it gets. The biggest problem I have with rain gear is getting too sweaty in temperatures above 50 F, but I've found the SP comfortable in temps up to 60 and higher so far since it is so well ventilated.

If you decide to try wool, I suggest a merino, or merino/synthetic blend.
Its much softer, & doesn't cause any itching for me.
Plain/heavy wool makes me itchy as well.

I actually have a few cashmere/silk V-necks I wear with collared shirts and bowties that don't make me itch. So maybe Merino would be OK.

If I comes across some, I'll check it out, but I'm quite happy with my current set-up.

The other (and for me main) advantage of all merino is it doesn't stink - EVER! When wet it sometimes gets a 'wooly' smell but merino is naturally anti bacterial so it never smells like me. In the summer I wear a light, short sleeve merino jersey every day, wash on the weekends (only because I think I should).

Second advantage - merino is the only fabric that retains heat when wet. I didn't believe this until I rotated between merino jersey layer, polartec, windbloc, etc and it's true. The merino can get soaked with sweat and still feels warm (though clammy). Can you tell I like merino?! My experience is that plain wool is good but merino is very different. I also stay away from merino blends. Downside - the stuff is expensive (but lasts forever).