Well, that's because EVERY time after you have sent out a byte, there will be a serial (tx) interrupt, no matter where you have come to in your main loop. And what does your serial ISR do on a tx interrupt? That's right: unconditionally send out a byte. It will go on forever.

A serial transmit ISR should check if there is something to be sent and send that, or else exit (almost) immediately. The rough idea is that all that checking if the transmitter is ready shouldn't be necessary in your main, and scanning the array and putting it in "outgoing" is exactly the sort of stuff that you normally put in your tx ISR.