How can I remember the SSH key password on a remote system that SSH's into a second remote system? I have three systems:

New MBP (Catalina)
Old MBP (El Capitan)
Raspberry Pi (Raspbian Buster)

I want to SSH from the New MBP to the Old MBP, and then SSH from the Old MBP to the Raspberry Pi.
My final goal is to set-up an rsync backup scheme between the Raspberry Pi and the Old MBP.
✅ I can SSH from the New MBP to the Old MBP without a problem.
✅ When I open screen sharing on the New MBP, connect to Old MBP and SSH from the Old MBP to the Raspberry Pi it works without a problem.
❌ When I SSH from the New MBP to the Old MBP, I can SSH from the Old MBP to the Raspberry Pi, but everytime it asks for a passphrase of the SSH key.
I followed these steps and found I had to set-up some different things as well on the Old MBP before they worked. So the complete things I did are:

SSH from New MBP to Old MBP
Start an SSH agent on the Old MBP: eval &quot;$(ssh-agent -s)&quot;
Add the key: ssh-add -K ~/.ssh/id_rsa and ssh-add -A
Now I can SSH to the Raspberry Pi without a password prompt: ssh user@raspi

As long as I don't logout of the Old MBP session (from step 1) everything is fine.
However, if I:

close the New-Old SSH connection
re-connect from the New MBP to the Old MBP
Try to SSH to the Raspberry Pi

I get prompted for a passphrase again!
How can I get the ssh-agent on the Old MBP remember the passphrase across different SSH sessions?
This answer provided me with a solution.
By using ssh -A user@new-mbp and then ssh user@raspi I re-use the ssh-agent of the New MBP (if I understand it correctly) on the Old MBP and can login on the Raspberry Pi using an SSH key without being prompted for a password.