Hi,
The simplest solution would be to have 2 set of shipping methods created in the System>Shipping methods menu. One set for the morning delivery and one set for the afternoon delivery.
That way, on the checkout, in the "shipping" view, instead of only one shipping method available for the current zone of the shipping address, the user would see two shipping methods, one for the morning, one for the afternoon and he would be able to choose between them.
And to know which delivery was selected, you can just look at the name of the shipping method. That way, no custom field would be necessary.