The Covenant as we knew it fell apart. The Prophets ordered the destruction of the Elites, the Brutes took the Elites’ place and the other species divided between the two sides (though it would seem the majority of militant units stayed on the Prophet’s side). All traces of the Prophets and most Engineers have vanished. Most of the Prophets are dead as they lived on High Charity which was overrun by the Flood. It’s possible that other Prophets are out there but no signs have been found.
So no, that Covenant wont come back. The reason Jul’s forces are referred to as “Covenant” is simply familiarity. They have no official name yet, Storm was a misunderstanding and remnant isn’t the most accurate term.
The Brutes divided into factions, some loyal to the Covenant religion, others against it, and have been waging their own war. Some surrendered to the Elites and served them for a time, but when some got together and launched insurgent attacks on Sanghelios, the Elites seemed to completely turn against them. There are some on Venezia, living side by side with humans and Jackals.
The Unggoy sided with whoever commanded them at the time, simply out of fear rather than any true loyalty. Some that fought under the Brutes were shown to support the Elites and Arbiter. This carried over into the post-war world were Grunts were sided with both Elites and Brutes. No transports a have left the Grunt’s home world, but there are Grunt populations in Elite systems, likely were those that serve them come from. It would seem that they are split between all sides.
After the fall of the Covenant the Jackals returned to their pirating ways. Some live on Venezia among human rebels and Brutes. As seen in Halo 4, some work for Jul’s forces. Whether they have any true loyalty or are simply paid mercenaries is unknown.
The Hunters split between the Covenant and Elite forces at the end of the war. Some know fight for Jul, it is unknown if they have all sided with him or are split like many other species.
We haven’t heard anything about the Drones at this stage.
The Elites are in the midst of a civil war. The major factions at this time seem to be the Arbiter’s forces (allied with the UNSC) and Jul’s forces which hold more religious beliefs and worship the Forerunner’s for the most part (some are simply against humanity and aren’t so devout). As SpOps Episode 1 indicated, there are enough ties between humanity and the Elites that Elite refugees have started living on Earth. Some Elite terrorists have taken advantage of this and used it as an opportunity to launch attacks.
So as you can see it’s quite complicated. Every species with a known status seems to be divided. There are still some who follow the Covenant beliefs from the original trilogy, but at this stage they seem too few to really cause problems.