As someone who frequently hunts down references, I can only offer a hit-and-miss approach. Until hadith collections and their translations are properly digitized and stored in a retrieval-friendly format, that's going to be the best we non-Arabic speakers can do. http://sunnah.com is one step in that direction but there is a long way to go.
The hit-and-miss approach is as follows:
- Find reliable websites that mention the hadith, say in English. You can use your paraphrasing to do this and most modern search engines will be able to figure out what you mean.
- Once you find a reliable website with the hadith translation, copy down the text of the translation (which you didn't have).
- Now search for this translation and see if you get any hits which include a reference. If so, track down that reference either on a website that has that collection digitized or in a PDF e-copy of that collection. If you found it, you're done. Typically however it's not that easy. Even if there is a reference, it typically won't be according to the standard numbering scheme. In that case, try to see what scheme they follow and where that hadith is in relation to the book and chapter numbers.
- Translate a few keywords of the translation into Arabic and force Google to search within websites that have digitized copies of the hadith collections (for example hadith.al-islam.com or al-eman.com or other websites). You will get multiple results and you will need rudimentary Arabic to sift through these. If you're lucky, they will be in the primary collections. Otherwise you'll get references, which you then track down.
Keep doing this and follow the leads and if it exists chances are you'll keep narrowing the field and eventually find it, inshaAllah.