When smith exploded because of neo he wasn't deleted or sent to the source, he was just killed, like agents have been killed before, like in the subway. After he was "exploded" by neo he naturally took over another body, yet now had the abillity to copy himself over people because of the way Neo killed him, and sort of rubbed off on him.
Shortly after he was terminated from being an agent, now an "exile" with the ability to copy himself. When he was terminated from being an agent, if he had been killed, it would have been the end of smith, because he would be unable to copy himself to another person. For example if smith wasn't an agent in the first movie, he would have died from being shot fromt he helicopter becasue he wouldn't be able to reposses anyone, or when he was smashed by the train, or exploded by neo, but he was an agent, so he repossessed someone each time.
However after being exploded he had the new ability to copy himself, so when he was removed from the agent network it didn't matter cuz he had multiple copies.
And which is also why he died in revolutions, because he wasn't an agent, and Neo had destroyed each and every smith inside the whole of the matrix. Hope that clears up things for some people.
PS: And if you were all wondering why he was glowing in the end, it was because he had absorbed every single smith so to speak, like he absorbed that sentinel, that is his ability, and how he destroyes the machines. He had cleansed everyone in the matrix, if they had cortex scarring they are now healed, and once again themselves. Saving the matrix, and when you saw all that energy that was the energy from absorbing all of the smiths, you even saw the energy of smith flow into him when he was crucified, dying to save everyone, killing him from the overdose. God it's so simple why can't people see these things the first time lol.