Is there a tool to dynamically redirect output to a new file on request

I'll suggest a named pipe.

1. Create a pipe mkfifo p (call it whatever you want, if not 'p')

2. Create a "reader" script that reads from the pipe and writes wherever you like

3. Tell the monitoring program to write its logs to the named pipe

Here's a sample reader script that reads from a named pipe 'p' and writes the data to an indexed 'mylog' file:

#!/bin/sh

INDEX=0

switchlog() {
  read INDEX < newindex
  echo now writing to "mylog.$INDEX"
}

trap switchlog USR1

while :
do
  cat p >> mylog."$INDEX"
done

Building up on your SIGINT idea, here using SIGQUIT (Ctrl+\) to you can still use Ctrl+C to stop the whole thing:

(trap '' QUIT; monitor_command) | (
   trap : QUIT
   ulimit -c 0 # prevent core dump so SIGQUIT behaves like SIGINT
               # for cat
   n=0; while n=$((n+1)); file=output.$n.log; do
     printf 'Outputting to "%s"\n' "$file"
     cat > "$file"
   done)

That assumes cat is not a builtin in your shell (so it does get interrupted by you press Ctrl+\).

Note that like in your approach, there's a chance that the SIGQUIT be delivered at the wrong time (in the write system call) causing some data to be lost.