In a cross-site scenario as yours, it may be preferable to use FQDNs to access hosts by name. Using plain, short names may produce unexpected resolution results if two hosts on each of your sites would use the same name (which would be true for pi.hole, e.g.).
If you still would want to supply multiple search domains, you could create a custom
dnsmasq configuration file (e.g.
/etc/dnsmasq.d/42-search-domains.conf) containing a line like:
Substitute each of
site2 with your respective search suffix as required; refer to
dnsmasq documentation for additonal details.
Then run a
dnsmasq syntax check:
If ok, restart Pi-hole:
Note however that there is no guarantee this would work:
Not every DHCP client may request and honor that option.
You may take care of the former by changing above line to:
This will make Pi-hole's DHCP server provide that option to a DHCP client, regardless whether it was requested or not. It won't do anything if a client does not digest it at all.
I'd recommend monitoring each of your clients for the desired behaviour.