Short explanation:
Yes, once you change the SSID all existing connections to that SSID get dropped and have to be reset with the new SSID and "reconnected" manually. This is because usually your your AP will drop all connections after making significant changes to the configuration. Although it depends on make and model - a different implementation will result in different behaviour: cheap devices (with multi SSID) will drop all connection for all SSIDs better ones will drop only the ones connecting to that specific SSID.
Long and detailed practical explanation:
A general/abstract explanation in layman's terms would go like this (Note: this depends on the actual implementation - and I never developed an AP and I just deduct this from my general development knowledge on how embedded system work in general):
The SSID will be saved somewhere on a persistent flash memory and that is what you access and overwrite when you change it. Re/Starting the AP will result in the value being "put" from flash into the AP's RAM from where it's being utilized for the AP's actual operation. This means changing the value during operation wouldn't do anything since you would just change the value on the flash memory which will not be used until the AP restarts. Of course in reality the AP "knows" that you changed the setting and takes steps to apply them. It will restart it's operation thus dropping all existing connections. Since "live updates" are more costly to implement and there isn't a use case to your scenario the programmer wouldn't bother changing the flash and RAM value when changing the settings. Why should he, when "restarting" has the same result (not restarting in a way you would through the GUI just restarting the "service" running on the AP).
Alternative answer with a long and detailed theoretical explanation:
Back to your question: Assuming your AP doesn't drop the connection actively when you change the SSID and just "uses" the new SSID instead of the old one, I think it could work but I'm not completely sure. So in this case no, you wouldn't have to reconnect manually.
Note, the following explanation is theoretical (and not within the scope of Superuser, since it's not about a "real" HW or SW problem):
The SSID is required for the authentication and association process. Once the session is completely established (not sure, but I think the 4 way handshake is required to start the session as well) I don't think the SSID is needed again. So as long as the session remains open you should be good to go.
However as far as I understand there is a timeout. If it gets triggered a new session will have to be established and the authentication and association process has to be repeated. Since you changed the SSID this will fail. Note: Restarting "WLAN restart" on the client or AP will close the session as well, but I think that's obvious.
Some further reading material about how APs work can be found here and here.
Addition: The SSID still can get transmitted e.g. within beacon frames (see Tim's answer). But at this point I haven't found any evidence that it does actually get evaluated the authentication and association process. The fact that it doesn't have to be transmitted within the beacon frame (it can be 0
, see here and here) leads me to believe that it does not get evaluated after the authentication and association process. But Superuser is not the right site to ask/answer this type of question anyway.