The thing you need is called a Wi-Fi Access Point (AP) - all Wi-Fi is facilitated by AP's, it's just that a lot of "other" things have AP's built into them such as SOHO routers, HomePlugs, etc.
If you have an old router lying around, you can repurpose that as an additional AP/switch combination. How to do so is described in the "Using Two Routers Together" FAQ pinned in this forum - there's a few hoops to jump through, but it isn't difficult. (You'll need an ethernet swtich in the mix either as a standalone device of as part of an AP/switch combo to avail connection of the backhaul/uplink, the AP and your TV.)
In "cellular" Wi-Fi deployments with multiple hotspots, the way to avail automatic "roaming" (as it's usually called) between hotspots is to ensure they all advertise the same SSID/passphrase. It's what we do on big sites with dozens/hundreds of AP's.
However, note that it is the client devices that decides if/when to institute a roaming assessment not "the system" thusly, clients are not (by design) constantly "hunting for the best signal" and some will doggedly hang on to a working link until it gets pretty grotty.
Some of the newer "whole home" and "mesh" systems aimed at the SOHO marketplace are beginning to implement (newer) technology to try and "encourage" clients to roam more readily, but it is by no means universally implemented at time of posting and even when it is, clients are not compelled to heed the "hint" from the AP's that they should roam somewhere else.
With multiple Wi-Fi cells, the coverage area will almost certainly overlap which means the hotspots could "interfere" with each other effecting performance. Thusly, it's best to ensure adjacent hotpots use different radio channels to prevent what we call "co-channel interference." Instead of letting the it auto-tune, for a small deployments of a few hotspots with a heterogeneous mix of kit, I'd tune the channel plan manually (most SOHO kit will let you choose channels.) For the 2.4GHz waveband, ensure the channels are a least "5 apart," so for example, pick two from the set [1,6,11] for the 5GHz waveband, just ensure they are different.