I use OS 1:25K maps in the UK. I remove the covers as soon as I purchase them and label each corner on both sides of the map with the map number with felt pen. I then refold them 1/2, 1/4/, 1/8, 1/16 and file them in my (cardboard) map box so one of the numbers is visible. (I do the same with my 1:50K maps).
When I'm preparing for a hike I refold the map, making new folds where necessary, so that the whole of my route and key landmarks can be seen without me having to unfold the map during the hike. The map then goes in the map case for protection from the wind, rain and chafe, and the map case is clipped to my pack so it can't blow away or get lost. It sits in the gap between my hip and the shoulder strap so it doesn't blow around in the wind like a neck-worn map case will.
I can recall rarely having to refold a map during a day hike. Even if you have to refold it on a day hike the map is only exposed to the weather once and for a short time. You can also mark the mapcase with chinagraph pencil if you need to mark key points on the map for your hike; this works fine assuming the map doesn't move around in the case! If you're using a GPS as well as the map, you can mark on the map (pencil or ink) the relevant KM square eastings and northings if they're not visible after you've refolded the map for your hike.
The advantage I see with having the whole map with you is that if you have to go "off piste" for some reason (bale out, medical assistance) you still have a map and a means of orienting yourself; this may not always apply if you just print out A4 sheets centred on your route.
I use one of
these map cases. I highly recommend them: the last one I had lasted me over 20 years and has only been retired as the plastic has fogged a little.
You may be interested to take a look at
this technique to cut and fold a map for special use.