Allison, Paycor has all those features and more, and I have found their solutions to be much easier to navigate, and their support structure a ton easier than say the #1 payroll provider on the tip of tongues.... you should ask your rep to set up a meeting to review their entire offering.

My company is with Kronos...one of those huge names in HRIS/Payroll systems.  Their system is high powered and very capable of handling even the most complex pay practices (if you need that sort of thing as my mind still spins when thinking about some of our pay practices).  We use the Kronos platform for scheduling, timekeeping, HR, and Payroll.  Beyond that, we have ADP cut our checks and pay our taxes.

We use ADP with Cincinnati Time as our time keeping interface.  We could use ADP's time keeping, but we are happy with Cinti Time and there is no problem with the interface.  We are happy with ADP and particularly how they helped us navigate ACA compliance this year.

We use Zenefits and like it - for all it's features. I've used or seen Namely and Bamboo HR as well. 

We too are small-to-mid-sized and are currently moving to PlanSource, which partners with Ultimate Software and other heavy hitters in the HRIS and Payroll arena.  During our RFP process and references, we found their customer service support (HUGE) and integration products top-notch. They will be helping with a lot - payroll, tax, HRIS, recruitment, performance, ACA, time/attendance, etc.  We are moving from ADP.