I'm running with Continental Contact MX5 235/50/R18 on original wheels currently, but after being poisoned badly enough by the look of the 20 inches, I thought about switching my wheels. But I'm still a little hesitant of moving up because of the sacrifice on the ride comfort.

I understand the bigger tyres naturally would cater more to handling performance rather than ride comfort, but is there any recommendation for 19 inches or 20 inches tyres that are all rounder - quiet and soft/comfortable as well as good in handling and doesn't wear out fast?

I'm looking for 245/45/R19 for 19 inches and either 245/40/R20 or 255/40/R20 for 20 inches.

I see. But what do you mean that it will 'rub a bit' when doing a maximum turn with the 245/45/19?? Do you mean that the profile will be a little oversized and that the tyres will be rubbing against the walls of the wheel opening?

Any recommendation for 245/40/20 tyres?

And do you happen to ferry old folks at the back seats complaining about bumpy ride on your 20 inches? I'm just trying to get some opinions and rough idea on the comfort level with 20 inches.

If 19' you should look for 235/45/19 to match your current total diameter so it would still fit perfectly. Comfort wise I'd say still  reasonable coz the sidewall height/thickness is still about 10.6cm however you will still feel significant reduction compared to your previous tire size since it's a drop of about 1.2cm tire sidewall height. Therefore based, I would also not recommend you to go 20' as you'd need to use even thinner tires (245/40/20 or 255/35/20) where the sidewall is already below 10cm ( drop 2cm or more from your current size) in order to maintain close to your original total diameter for fitment clearance.

You can do tire size comparison here:
http://www.1010tires.com/Tools/Tire-Size-Calculator
Well 245/45/19 and 245/40/20 is practically the exact same total diameter with only about 1mm difference. So the cause for your rubbing should not be because of the tire size but due to the rim's offset & width where the 19' rims you chose that time did not have the proper size to fit your car.

You mentioned going from 19 to 20 inch rims so that's already different rim sizes which means can also have different widths and offsets. If your 19' that time was maybe wider and/or offsets too out or too in, then it might rub something. If both your rims have exactly the same widths & offsets just different diameters (19 vs 20) then it should not rub because the total diameter (including the tire sizes that you use) are about the same.

If 245/40/19 vs 245/45/19 then of course there's a significant difference (1.2cm sidewall height difference or total 2.4cm difference in diameter) so it makes sense that 245/40/19 should not rub.
But 245/40/20 vs 245/45/19 is only 1mm difference in total diameter which practically the same so it should be the same, provided the rims width and offsets are the same or very close. Now your example above shows your 19' offset is quite a lot different than the 20'. Now even though your 19' width is a bit narrower, but the effective tire tread width is still the same 24.5cm and mounted on a rims which is 4mm more outward, so that might make it rub. If you use 19x8.5 offset also 42 then it should not rub with 245/45/19.