tedding a couple of hours after cutting will have the best reduction in drying time.

The secret of drying hay is to have as much sun shining on as much hay as possible. The sun warms the juices creating a raised vapor pressure, forcing moisture out the nearest opening. 
Next there are openings in the leaves and stems (Stomata) that will close if they are in the dark. i.e. hay that is inside a windrow, or hay that is still alive after sunset.
HERE the leaves will be dried down to below 42% moisture in about 5 hours of good sunshine.
The leaves will rehydrated with the night humidity, but that will quickly go away with dew burn off. That means only the stems will have much moisture to need loosing during the second or later days.

I agree it is extra work and equipment to run a tedder to spread out the hay.  If you use a 12 swather the widest the swath can be dropped is maybe 6 feet, but using a tedder on this it is easier to spread than a damp windrow.

With NH and some other drag behind disk conditioners it is possible to have the hay covering close to 100% of the ground with some minor modifications of the machine.  NH sells a kit. For my OLD NH 411 it was a little more than removing the doors and bolting on the fins. But it works.  I have not used a tedder for two years now.
 

It is a lot fun to use a tedder at 540 PTO rpm's. Fun to throw out a rooster tail but it will encourage extra leaf shattering.  HERE bermudagrass hay is worse than alfalfa for leaf shatter.  I always have to blow more leaf stuff off my baler with bermudagrass than with alfalfa. My barn floor is the same with more grass leaves than alfalfa leaves. 
I have not baled any tall grasses of late and can not say they hold leaves better.

Our learned University Types tell us to use a tedder whit the hay's moisture above 50%, and to rake with the hay's moisture above 40%.
When cut the moisture is in the 80% range. But the next day when the humidity is 90% THEN the hay moisture will be no LOWER than 40%! 

I have noticed that bermudagrass hay, with most of the leaves knocked off, will still look good. For alfalfa with most of the leaves gone, will look like a bundle of sticks. Maybe green sticks but still look very stemmy. 

Using the humidity to our advantage it is possible to have the stems dried to close to 10% moisture and the leaves still at 30% moisture to save the leaves and not have mold or heating.  Reverse that and have a bundle of moldy stems.

In our Arid West they bale at night to find enough humidity to hold the leaves. Some spray the hay with some water, some inject water into the windrow, and the ultimate, For large Bales, they inject steam into the windrow. 

The real deal on humidity for baling is fairly universal, but the timing is different for each climate.