Integrated amp with phone and headphone jack?

Whoops, I meant "phono" not 'phone.'

This would be for a second system.

anybody?

Creek Classic 5350SE
Arcam A90

both have headphone jack standard and optional phono boards

Yes, and more times than not, it is called a receiver, and includes a tuner. The Marantz solid state receivers of the 1970s are excellent integrateds, and the tuner section is also very fine by today's standards.