format_exc() is really just

etype, value, tb = sys.exc_info()
return ''.join(format_exception(etype, value, tb, limit))

So if you have the exception type, value, and traceback ready, it should be easy. If you have just the exception, notice that format_exception() is essentially:

a_list = ['Traceback (most recent call last):\n']
a_list = a_list + format_tb(tb, limit)

where limit defaults to None.

Example:

To further aid in clarifying a (simple) application of format_exc():

import traceback

try:
    # Some code
except Exception:
    print('\nThe following error occurred:\n',
          traceback.format_exc(),
          'Processing aborted.\n',
          sep='\n')
    # sys.exit(1)

The answer to this question depends on the version of Python you're using.

In Python 3

It's simple: exceptions come equipped with a __traceback__ attribute that contains the traceback. This attribute is also writable, and can be conveniently set using the with_traceback method of exceptions:

raise Exception("foo occurred").with_traceback(tracebackobj)

These features are minimally described as part of the raise documentation.

All credit for this part of the answer should go to Vyctor, who first posted this information. I'm including it here only because this answer is stuck at the top, and Python 3 is becoming more common.

In Python 2

It's annoyingly complex. The trouble with tracebacks is that they have references to stack frames, and stack frames have references to the tracebacks that have references to stack frames that have references to... you get the idea. This causes problems for the garbage collector. (Thanks to ecatmur for first pointing this out.)

The nice way of solving this would be to surgically break the cycle after leaving the except clause, which is what Python 3 does. The Python 2 solution is much uglier: you are provided with an ad-hoc function,sys.exc_info(), which only works inside the except clause. It returns a tuple containing the exception, the exception type, and the traceback for whatever exception is currently being handled.

So if you are inside the except clause, you can use the output of sys.exc_info() along with the traceback module to do various useful things:

>>> import sys, traceback
>>> def raise_exception():
...     try:
...         raise Exception
...     except Exception:
...         ex_type, ex, tb = sys.exc_info()
...         traceback.print_tb(tb)
...     finally:
...         del tb
... 
>>> raise_exception()
  File "<stdin>", line 3, in raise_exception

But as your edit indicates, you're trying to get the traceback that would have been printed if your exception had not been handled, after it has already been handled. That's a much harder question. Unfortunately, sys.exc_info returns (None, None, None) when no exception is being handled. Other related sys attributes don't help either. sys.exc_traceback is deprecated and undefined when no exception is being handled; sys.last_traceback seems perfect, but it appears only to be defined during interactive sessions.

If you can control how the exception is raised, you might be able to use inspect and a custom exception to store some of the information. But I'm not entirely sure how that would work.

To tell the truth, catching and returning an exception is kind of an unusual thing to do. This might be a sign that you need to refactor anyway.

Since Python 3.7 you can create traceback objects dynamically from Python.
To create traceback identical to one created by raise:

raise Exception()

use this:

import sys
import types

def exception_with_traceback(message):
    tb = None
    depth = 0
    while True:
        try:
            frame = sys._getframe(depth)
            depth += 1
        except ValueError as exc:
            break

        tb = types.TracebackType(tb, frame, frame.f_lasti, frame.f_lineno)

    return Exception(message).with_traceback(tb)

Relevant documentation is here:

• https://docs.python.org/3/library/types.html#types.TracebackType
• https://docs.python.org/3/reference/datamodel.html#traceback-objects
• https://docs.python.org/3/library/sys.html#sys._getframe