This is usually configured on the host where you're running your browser, and it depends on the operating system of that host.
For example, on a Mac you would go to System Preferences/Network/Advanced.../DNS, and add
myhomenetwork.com to Search Domains in the right pane.
On a Ubuntu linux system, you would edit /etc/network/interfaces and add the line
[User duct_tape_coder added the following:]
On Windows, edit the adapter settings, select the IPv4 or IPv6 properties, click Advanced, click DNS and change the radio option to "Append these DNS Suffixes (in order)". By default the system will only apply the Primary DNS suffix (set on the same page you set the computer name). If you haven't joined AD, the Primary DNS suffix is likely empty.
In CentOS7/RHEL with Network Manager, use NMTUI to edit the NIC and add the search domains:
A reboot of the computer should cause the change to take effect but may not be necessary.