I own and shoot ( and occasionally compete) with three .223 match rifles as well as several others.
The .223 can reach out there, for sure, but stacked up against other cartridges, it is left way behind in the dust past 600 meters , especially if you are talking about off the shelf rifles.
Almost anything with decent optics in 6mm and up will best it.
The 6.5 Creedmoor is offered in a host of factory rifles these days and is a good alternative.
I didn't notice on whether or not you are hand loading. If not, and you intend to continue on a regular basis, hand loading is essential to success at extended ranges.
Cat
Agree, I do not compete but own a .223 target rig and I can stretch to 600 on the outside at best. I know of some amazing competitive shooters who can take it to 800 but man, that's a looong shot with a pipsqueak. And that'll be with a 24-26" barrel, highly tailored and very stout loads, 75gr+ projectiles which might mean custom throats, etc.
6mm is almost a cheat code for making those shots and I'm finding a surprising amount of 6mm Creedmoor on the shelves locally. But not a lot of good rifles available from the factory.
6.5 Creedmoor might kick harder but the 140+gr bullets just go. Ammo is easy to find, they don't seem to be too particular with bullets, the cartridge specs call for tight necks as well as throats designed for long bullets and fast twist rates to stabilize, they fit easily into a short action and standard mags, velocities are reasonable leading to good barrel life, easy to reload for, doesn't need a ton of powder. I know it's popular to hate on the Creed but man does it do a good job of making an okay shooter look like a hero. Don't even need to brake one if it's a target rifle, a 10lb Bergara B14 HMR with another 2 pounds of scope, rail, and rings on it plus a good bipod will be more of an angry shake than a hard kick.
People are taking 6.5CM out to a mile, so I'd probably start with that and find my limit from there.