A different problem than it looks
Your main room is full. There is a hall, a chapel, a classroom next door with seats in it, and a screen or a projector already sitting there.
The instinct is to point that screen at your online stream. It is already running, so why not?
Because the online stream is built for a completely different job, and using it in the building next door produces three problems you do not have to accept.
Why the online stream is wrong for the next room
The delay is severe. Streaming platforms buffer heavily... that is how they deliver reliably to thousands of people on unpredictable connections. The cost is latency, typically many seconds and sometimes much more.
For a viewer at home, that is invisible. For a room thirty feet from the main hall, it is a problem you can hear. The overflow room applauds late. It laughs at a joke the main room finished with. If both rooms can hear each other at all, the gap is uncanny.
It depends on your internet. The overflow room is inside your building, connected to your network, a short distance from the source. Routing that signal out to the internet and back means a hall on the other side of a wall goes dark when your connection wobbles.
It costs upload bandwidth. Every screen watching your online stream consumes capacity that your remote viewers need. Serving your own building through the internet is paying twice for something you already have.
Send it across your own network instead
The alternative is to keep the signal inside the building.
The service goes from your production machine to screens in other rooms over your own network. It never touches the internet. The consequences are all favourable:
It is fast. Local delivery avoids the buffering that platforms add for global distribution. The next room stays close enough to the main room that the difference stops being distracting.
It does not depend on your internet connection. Your overflow room keeps working during an outage, because nothing it needs leaves the building.
It costs no upload capacity. Local traffic stays local. Your online viewers are unaffected no matter how many rooms you add.
Usually it needs no new hardware. If the screen in that room can open a web browser (a smart TV, a laptop, a tablet, an inexpensive stick) it can be the receiver. There is nothing to install.
More than one kind of second screen
Once the service is available across your building, "the overflow room" turns out to be one case among several.
An overflow room gets the full program: exactly what the main room sees.
A lobby or foyer screen shows the same thing for people arriving or moving about.
A crying room, a parents' room, a creche... anywhere people need to be present but separate.
Personal devices. People can watch on their own phones on the venue network. Useful for someone with a restricted view, a carer in a side room, or an usher who needs to know where the service has got to.
A lyrics-only view. Rather than the full video, a screen showing just the current line. Useful for a backup vocalist, a room where the words matter more than the picture, or a display too small to make a video worthwhile.
Those last two are the ones venues do not think of in advance and then use constantly.
What this needs
A network that reaches the rooms. The main requirement. If your wireless does not cover the far hall, that is the problem to solve... and solving it is useful for everything else too.
A screen that opens a browser in each room. Smart TV, laptop, tablet, or a small stick behind a monitor.
A capable enough production machine. Every additional output is more work for the laptop. This is a real cost, and it is where a modest machine starts to struggle if you are also streaming and recording.
Worth being clear about tiers: multi-room casting is a feature of the higher tier, not the free one. If you are running the free tier, the projector output still covers a second display in the same room... the network casting to other rooms is a paid capability.
Practical notes
Check the delay yourself. Stand where you can see both rooms and watch the gap. Local delivery is dramatically better than an internet stream, but "instant" is not a claim anyone should make. Know your actual number so you can decide whether it matters.
Consider the audio. Video in the overflow room is straightforward; audio needs thought. If the rooms share a wall, sound bleed between them can be worse than either room alone. Sometimes the answer is a physical audio feed rather than the one attached to the video.
Protect the pages if the network is public. If visitors can join your Wi-Fi, they can reach anything on it. There is a password option for the viewer pages, and it is worth using... not because anyone is attacking you, but because a service should not be casually accessible to whoever is parked outside.
Test with the rooms full. A network with two devices behaves nothing like a network with two hundred phones in pockets. Your quiet-building test is not the test.
The summary
If the screen is in your building, do not send the signal out of your building.
The internet stream is for people who are not there. For the room next door, going over your own network is faster, more reliable, cheaper, and usually needs nothing you do not already own.
It is one of the few upgrades where the better answer is also the simpler one.