Integrated with good headphone output

The Onkyo A-9555 is a very fine integrated you could find used or refurbished in your price range and it has a good headphone output. There's a full review of the amp on the Stereophile site.

Another option for greater versatility than a dedicated headphone amp would be one of the many DAC's that include a headphone stage. Both the Beresford and Aune DAC's I'm using now have fine headphone output, as did the more expensive Benchmark I had before.

I can't give any recommendations for an integrated amp, but there are a lot that do take the headphones off of the speakers outputs, that is amplified through the whole amp. Without knowing if this is done from someone that has a schematic, or manufacturers provided info, there might be one way to tell. If the amp has a preamp amp loop with jumpers, if you take them out (jumpers) and the headphone jack keeps working, that should tell you it has its own amp stage, or at least is not going through the power amp too. If the jack shuts off, it obviously is going through the whole output stage also.

Willing to try some DIY?

Check out the CMOY pages. Most of what you'll need is at the RadioShack. The rest? A single order to any one of the electronic vendors. Than you can get into 'opamp rolling'. Buying the tools, for those without tools may be expensive, but you'll never have to buy 'em again.

You can build one for 30$ or 40$. It should drive the Grado phones just fine.

http://tangentsoft.net/audio/cmoy-tutorial/