A standard clarinet should be fine, and there are many very good second hand instruments in your 1K price range. Less risky is going to a store that has second hand stock for you to try a few, assuming as a beginner / intermediate, you already play to some extent. You may not get the bargain you might find on Ebay, but a good store with stock should have checked the instrument over by a technician. You should therefore avoid the shock of condition and repairs needed from an Ebay purchase. Which brings me to the point I am typing to make!
To respond to your comment about long thin fingers, you will notice on a clarinet that the upper and lower joint both have open tone holes, that when you cover the tone hole with your finger, a ring is pushed down.
Those rings close pads on tone holes higher up the instrument, and whether the rings are actuated enough to close these auxiliary tone holes depends on individuals' fingers. A thin finger may not push the ring down as far as a fatter finger. This is all no problem if the clarinet is adjusted to your finger, which is easily achieved by bending the keys as needed to correct the timing of the auxillay pad covering its tone hole at exactly the same time the finger pulp covers its open tone hole.
A good shop with a technician would be able to make sure adjustments that are needed for each player are attended to. Everybody's hands are different, and anyone who gets a clarinet new to them should be aware that trill keys, left little finger touch piece height, and ring keys are all fair game for adjusting to an individual for best response.
Chris