The idea that a Martini originated from a Martinez is somewhat disputed and I'd say completely false. A martinez is basically a gin Manhatten, martini's are really quite different drink and there's not really anything in the ingrediants or flavour profile to suggest they're related. That said, I do always keep a bottle of old tom gin (usually haymans) in the liquor cabinet for when i feel like one.

Negroni has been mentioned but as the greatest cocktail in the world it deserves another mention. My two comments would be rather than using the classic 1:1:1 recipe I'd suggest you overpour the gin a tiny bit so the ration is more like 4:3:3. I'd also say that they work best with a juniper forward gin which neither the Bombay or Hendricks are (Tanqueray would be the value for money gin here, my favourite in Negroni's is Four pillars)

Eastside - This is a Hendriks cocktail and variation of what you're drinking now but a bit more interesting. Basically muddle 2 cucumber slices and 2 mint leaves, add 2 Hendricks, 1 simp syrup and 0.75 lime. Shake and strain into ice filled glass, top with soda. Ellison is a slight variation using some bitters and served up rather than over ice/soda

Corpse reviver #2 - One of my favourites and has a kick. This does work very well with Bombay, it does have a couple of less common ingrediants but well worth the effort

Hanky panky - Similar to the aformentioned Martinez but with more herbal notes, nice after dinner

Bijou - Classic new orleans cocktail, a bit boozy, might need a a bit more juniper led gin

Last Word - Popping up on lots of cocktail lists now. A bit lighter/more delicate in taste than a lot of the mentioned cocktails and would work well with either bombay or hendricks

Singapore Sling - Great cocktail for a sunny afternoon sitting on the balcony. Signature cocktail of Raffles hotel and this is a travel forum

Southside - simple cocktail that doesn't need many other other alcoholic ingredients, should work with Bombay and I think is in the ballpark of what the OP would like.

Basil smash - this is getting quite popular these days, again a refreshing afternoon drink I suspect the OP would like using some muddled basil leaves for flavour. You can play with the ratios in this a bit to get what you'd like.