Yes that is how it will work when you have different SSID. The other option is to use the same BUT then you have the other problem that the end device stays connected to the first router and does not switch even though the second connection is stronger. The end device does not have the ability to scan for a second signal when it is using its radio to talk to the first.
There is no really good solution to and is mostly a personal preference.
There is talk they may release a newer wifi standard that has more control for the roaming placed in the routers rather than the end devices. Even if they do it likely means replacing every device so it will take a long time to become common.