You could do this with sed:
Example 'one' script, emits one line per second to show it's line buffered and running.
#!/bin/bash
while [ 1 ]; do
echo "TICK $(date)"
sleep 1
done
Then pipe that through this sed command, note that for your specific example 'ArbitraryText' will be the number of fields. I used ArbitraryText so that it's obvious that this is the inserted text. On OSX, -l is unbuffered with GNU Sed I believe it's -u
$ ./one | sed -l '1i\
> ArbitraryText
> '
What this does is it instructs sed to insert one line before processing the rest of your file, everything else will pass through untouched.
The end result is processed line-by-line without chunk buffering (or, waiting for the input script to finish)
ArbitraryText
TICK Fri Jun 28 13:26:56 PDT 2013
...etc
You should be able to then pipe that into 'two' as you would normally.