La commande tail -f est faite pour ça

-f Follow option. If the input file is a regular
file or if file specifies a FIFO, do not terminate
after the last line of the input file has been
copied, but read and copy further bytes from the
input file when they become available (tail enters
an endless loop wherein it sleeps for one second
then attempts to read and copy further records
from the input file). This is useful when
monitoring text being written to a file by another
process. If no file argument is specified and the
input is a pipe (FIFO), the -f option is ignored.