Rhaegar thought that Aegon was the 'Song of Ice and Fire', as we saw in ACoK in the vision Dany has in the House of the Undying. Rhaegar apparently believed that Aegon was the Prince Who Was Promised until he died. We can assume thus that in his view and possibly Elia's, who was also in the same vision, Aegon's survival was for more important than their own for the future salvation of the realm (since they believed that Aegon would save the realm from the Others). With King's Landing in danger, it seems possible that Elia and her servants would have concocted a plan to save Aegon.
Assuming a servant did sneak the baby out of the city, where would they take it? The other Targaryens were on Dragonstone, which was under blockade by Stannis and Paxter Redwyne's ships within weeks of the Battle of the Trident. Obviously staying in the Seven Kingdoms was highly dangerous. What would be needed would be a friendly refuge on the eastern continent. However, Elia would not have known who to trust and who not to. She'd have placed her trust in one of Rhaegar's closest friends and, hey presto!, sitting in the Free Cities seriously annoyed at being exiled for no good reason we have Rhaegar's friend Jon Connington (and former Hand of the King who was exiled after losing the Battle of the Bells, referenced in ASoS). Baby Aegon could have been sent into exile with Connington.
This is backed up by Tyrion's second chapter in ADWD (read out at cons by GRRM) in which Tyrion learns from Ilyrio that he is going to meet 'Griff', the head of a mercenary company, and his 'son' 'Young Griff', who seems to be held in special importance. And the symbol of House Connington is a griffon.