mdf: USB - C to HDMI mi adapter works well with modern Samsungs IME.

And the S10e does support HDMI Alt mode, so a USB-C to HDMI connector will work, without the need for a travel router - see
https://www.samsung.com/au/support/mobile-devices/how-do-i-connect-my-phone-to-my-tv-using-hdmi/ 

mudguard:

 

I think I'm in motels/hotels seven weeks out of eight. If I was in the same place each night I'd look for a solution for a bigger screen, but honestly taking more equipment with me means I'm liable to leave it plugged in and forgotten. 

 

I just use my laptop if I really want to watch something. The wifi can be completely hit and miss so I wouldn't want to rely on it. 

 

I've found windows netflix app helpful in this case as it allowed me to download my content beforehand (cannot do that on the browser).

dafman:

 

Any recommendations for a good travel router?

 

Depends if you also want 3G/4G.

I have a Hootoo filehub.  It connects via WiFi or ethernet also sas a power bank and you can plug USB HDD USB SSD and USB sticks into it.  [Edit: Great No Mobile data capability though].

PBTech sell a TP Link model that does 3G/4G.  I've no experience of it.