Use the disown shell builtin to exit the shell but continue running already back grounded jobs. The syntax changes from shell to shell.

zsh

disown [ job ... ]

bash

disown [-ar] [-h] [jobspec ...]

Example:

foo &
disown %1
exit

Relevant commands:

• find the parent pid

ps xao pid,ppid,comm | grep foo

• get information on a process

ps aux | grep NNNN

Ref:-

• https://www.slashroot.in/disown-command-linux-explained-example-usage - goes into the details; good explanation with relevant screenshots; has some typos;
• relevant section in zsh documentation: http://zsh.sourceforge.net/Doc/Release/Shell-Builtin-Commands.html → search for disown
• relevant section in bash documentation: http://manpages.ubuntu.com/manpages/bionic/man7/bash-builtins.7.html → search for disown → go to the second hit

Note:- To attach a disowned job, the instructions in https://unix.stackexchange.com/questions/31824/how-do-i-attach-a-terminal-to-a-detached-process might be helpful. But I have not tried them.

tags | continue running the jobs in the background, detach