I found some sentence in the online newspaper;
The Chilean flamingo chicks started to hatch on July 18 with 13 more arriving since then.
Does it mean some chicks arrived on the 18th and after they arrived 13 more chicks have arrived?
The sentence, as it is written, means exactly what you say. However, it seems to me that the sentence is rather clumsy. What it says is that some unknown number of checks arrived, and then 13 more arrived. That's a strange thing to say - one unknown number combined with one highly precise number, which is pointless since we don't know the exact total number of chicks, which would be the interesting number.
I wouldn't be surprised at all if the actual facts were different from what the sentence says, and that the sentence was just clumsily constructed.
According to Seaworld, of all places, Flamingo hatching can take up to 36 hours, making it quite likely that some chicks arrived on the 18th and more followed them afterwards. So yes, your interpretation is probably correct.