Get next enumerator constant/property

Absolutely.

Just add the desired functionality to your Days class:

class Days(Enum):

    Sunday = 'S'
    Monday = 'M'
    Tuesday = 'T'
    Wednesday = 'W'
    Thursday = 'Th'
    Friday = 'F'
    Saturday = 'Sa'

    def next(self):
        cls = self.__class__
        members = list(cls)
        index = members.index(self) + 1
        if index >= len(members):
            index = 0
        return members[index]

and in use:

today = Days.Wednesday
print(today.next())
# Days.Thursday

---

While the above is probably easier to understand, it is possible to do the work once in __init__ by adding a next attribute to each member (and previous while we're at it).

class Days(Enum):
    #
    Sunday = 'S'
    Monday = 'M'
    Tuesday = 'T'
    Wednesday = 'W'
    Thursday = 'Th'
    Friday = 'F'
    Saturday = 'Sa'
    #
    def __init__(self, value):
        if len(self.__class__):
            # make links
            all = list(self.__class__)
            first, previous = all[0], all[-1]
            previous.next = self
            self.previous = previous
            self.next = first

and in use:

>>> Days.Tuesday.next
<Days.Wednesday: 'W'>

>>> Days.Tuesday.previous
<Days.Monday: 'M'>

>>> Days.Saturday.next
<Days.Sunday: 'S'>

>>> Days.Saturday.previous
<Days.Friday: 'F'>

NB Using the this method of attributes means we no longer need the ()s after next/previous.

You can create a dictionary to lookup the next day like so:

In [10]: class Days(Enum):
    Sun = 'Su'
    Mon = 'M'
    Tue = 'Tu'
    Wed = 'W'
    Thu = 'Th'
    Fri = 'F'
    Sat = 'Sa'

In [11]: days = list(Days)

In [12]: nxt = dict((day, days[(i+1) % len(days)]) for i, day in enumerate(days))

Quick test:

In [13]: nxt[Days.Tue]
Out[13]: <Days.Wed: 'W'>

In [14]: nxt[Days.Sat]
Out[14]: <Days.Sun: 'Su'>