I periodically hit upon an out-of-the-box source for information about Jews prior to census records. One of these are the Navy Lists put out by the British Admiralty, some of which can be found on Google Books. The Navy licensed individuals to act as agents in port cities and most of those for petty officers and seamen were Jews. (I've found only one Jewish name for those acting for the higher ranks.)
The lists include the date the individual was licensed and the licenses were good for three years. This means that a directory published for the year 1820 (correct up to Dec. 1819) may include people licensed in 1816.
So far, the earliest directories I've found with lists of agents are for 1816 and 1820. There are also ones for the 1840s and later.
Sometimes there are clues to possible family relationships. For example, I've found a couple of entries where individuals are entered as executors for a deceased agent. Executors were usually sons, brothers, or sons-in-law, so where a surname was different from the deceased, this person may have married into the family. (I know this to be true for one case as the deceased left a lengthy will.)
I would like to know of other random sources people have discovered. (Old Bailey records are also helpful, but they can give a rather jaundiced portrayal of our forebears.)