Given the PID of the first process, the loop
while ps -p $PID; do sleep 1; done ; script2
should do the trick. This is a little more stable than pgrep and process names.
Maybe you can press ctrl+z first and enter
fg; echo "first job finished"
Polling is probably the way to go, but it doesn't have to be horrible.
pid=$(ps -opid= -C your_script_name)while [ -d /proc/$pid ] ; do sleep 1done && ./your_other_script