Couple of suggestions
http://www.brookgreenhotel.co.uk/
Really nice gastro type pub – Google says its a 12 mins walk from Hammersmith
http://www.oldshipw6.com/
This is a nice pub by the river – Probably a 10 – 15 min walk – Don’t follow the map on google as that takes you a long the road in a weird route for 33 mins!
http://www.thehammersmithram.com/
Part of a small chain – Haven’t been to this one but been to others and haven’t been disapointed – about a 10 min walk
That should get you going – There are plenty more suggestions if you need them