Such an event within an already existing universe could have happened in two different ways:
1. The remainder of the previous universe is a black hole singularity without a size. This wouldn't be any different from the standard Big Bang paradigm.
2. The recent universe still had some measurable size before the second Big Bang happended. In order to be consistent with the observations, a Big Bang within an already existing universe with a size larger than zero would have had the following properties:
a) hot explosion
Reason: The cosmic microwave background (CMB) tells us that the universe must have been very hot at some time, even so hot that all matter was ionised and the universe was fully opaque.
b) liberation of "dark energy" to restart expansion
Reason: Current research attributes the expansion of the universe the properties of the miraculous "Dark Energy" that has a repelling character. It is NOT driven by an explosion. The "explosion" is the visible effect of the driving force.
c) had to happen everywhere at the same time
Reason: The CMB is fairly isotropic. If it had happened at one single spot are in a confined area, the CMB should possess a clear gradient.
d) a trigger with an infinite velocity of interaction
Reason: Anything that is able to cause an event across a non-zero extent must interact instantly everywhere. This implies that the propagation of such a cause must have an infinite velocity which contradicts everything we know today.