It depends a lot on how your partner feels about it.
However, I would say that an engagement ring symbolises a promise between two people - two specific people. It's sort of a symbol of the promises made between your husband and you. Therefore, I would find it very odd to simply "re-use" it with a new partner, and - apologies - disrespectful.
Then again, to some people the ring is just a ring, without any special connotations.
Could you wear the ring on your other hand? Or have the stone set into a pendant so you can wear it close to your heart without the hassle of another ring?
As for engagement rings, do you actually want another ring? Could you buy a flashier wedding band in lieu of an engagement ring? Or have an eternity ring?
Maybe I am the odd one, but I would find it pretty odd to choose exactly the same ring as well! If you like the look of solitaires, why not consider one in ruby, or sapphire, or emerald, etc?