I always liked this card and its mechanics and it is the signature card for this deck.
Most common way to utilize it is Nevinyrral's Disk. After Disk resolved, the board is clear to get damage in with the hatched Bird. Disk is the best option because it removes everything that could hinder the Bird (except lands, specifically Maze of Ith).
I wanted a different route for this deck.
First thing to consider is how you are going to put the Egg into the graveyard (from play!) to get the Bird. In order to be competetive in comparison to the Disk concept, this process needs to be beneficial and work in a direct way, so not by using another card to combo. Direct damage like Lightning Bolt and also Terror are being ruled out. Finally, I came up with the following cards, described in detail:
Diamond Valley
Allows sacrifice and even for life equal to toughness. Furthmore, it is a land that cannot be countered as it is brought into play. Defintely up for consideration.
The Tabernacle at Pendrell Vale
Quite an interesting one: You just don´t pay the upkeep cost and the Egg is gone. Disadvantage would be that the Bird is now affected by the upkeep cost too. Also not good if the deck is going to be creature heavy.
City in a Bottle
The only artifact in this list. It can work under some circumstances but it could prohibit the Egg from being played. Mentionable but not up for consideration at this point.
Balance
Arguably the best card that white has in this format so always a card to be considered. It is situational but could make you come out on top after sacrificing the Egg.
Wrath of God
Well, all creatures gone and just Birds left. Perfect situation I´d say but there might be rare cases where the Birds still do not manage to get in. Up for consideration.
Earthquake
Good option for direct damage and creature destruction but only for non-flying creatures. There will be quite some times where the opponent still has flying creatures to deal with the Bird so this card will not be in the upper ranks. At least it is red..
Inferno
Another good red option for direct damage and creature destruction but the mana cost is very high. Best case would be to get Birds out before having 7 mana available. Not my go-to card.
Fallen Angel
Sacrifice to boost up the Angel sounds good. 3/3 is not that strong and it can be killed by the common Lightning Bolts and Chain Lightnings. Overall a good choice though so up for consideration.
Gate to Phyrexia
Very interesting sacrifice option because you can destroy artifacts with it. Actually one of the few options for black to deal with artifacts and even for low mana. Also up for consideration.
Hell's Caretaker
It can sacrifice the Egg. Also has the potential to exchange small creatures for bigger ones out of the graveyard. Unfortunately, there needs to be at least one creature in the graveyard for this to get going. Let´s keep it up for consideration.
Hellfire
Mass creature destruction but too many downsides: High mana cost, only works for non-black creatures and deals a lot of damage to yourself potentially.
Lord of the Pit
Iconic and powerful creature and one downside could be used an advantage in this case. The other downside is very high mana cost. But for that mana you really get one of the biggest creatures available. To be considered for now.
Pestilence
This card could excel in certain decks that manage to synergize with it. Medium mana cost and you would need to pump further (black!) mana into it but it could have long term use. Not the direction that I want to take for now but it needs to be considered.
Sacrifice
Sacrificing for black mana that can be used for (bigger) creatures, X-spells and so on. Definetly up for consideration.
Season of the Witch
Very interesting mechanic. It forces attacks and Egg can be destroyed by just choosing not to attack. Forced attack can of course be a downside to yourself and it also costs 2 life per turn. Not really a go-to choice for now but notable card.
The Abyss
Very strong card that can be used to destroy the Egg but it is also a threat to the hatched Bird. Not really what we are looking for.
Berserk
This card can win games but the Egg has 0 power so we would just destroy the Egg without other benefits. Does not really fit our needs.
Cyclone
Potential mass creature destruction and direct damage but very slow and mana intensive. Excluded for now.
Drop of Honey
Similar to The Abyss, this card provides great creature destruction but does not fit because it would kill the Bird at some point.
Metamorphosis
The green version of Sacrifice. Noticable is that it provides any mana but it may only be used for creatures. Good option overall.
Volcanic Eruption
The only blue card in this list. Very bad for our purpose because it only works if there are mountains and it would cost at least 6 mana.
