If you keep finches, and keep birds that are compatible to each other and have a large enough cage or flight to accommodate several birds, it is often possible to add one more finch.
Before thinking of adding any more finches, you must make certain that all of the above conditions are met.
You do not want to add a finch that is aggressive, to a flight containing waxbills or similar finches.
You do not want to add another finch to an already overcrowded cage.
You do not want to add a tiny waxbill finch to an flight containing budgies or lovebirds.
One of the first things you should do is check out a finch compatability list. You can find one at Finch Niche Compatibility Chart and another at Finch Information Center. These show several of the most common finches kept as pets and it shows compatibility for breeding purposes as well as non breeding purposes (for instance if you just kept all male finches).
Any new bird must be kept in quarantine for a minimum of 45 days to protect your current flock as well as to protect the new finch. After quarantine, there is normally no problem in just adding the new finch or finches directly into the flight or large cage. There will, of course, be some introductions and curious onlookers, but finches seldom need any time to get used to their new surroundings and new flock members.
You should have plenty of perches to accommodate the new finches and plenty of flight room for all the birds. You may want to put an extra food and water dish on the ground for the newcomers until they find the main food supply. Even with compatible birds, there often is a need for more than one food and water supply at all times.
I find that it is very relaxing to watch busy little finches flying and chattering and just being finches.