|2d6||Research, Training, or Investigation Results (monthly)|
|2||Injury or Catastrophe (1d6-2 damage or cost multiplier)|
|3-5||Lost Time, start over (reroll cost)|
|6-8||Everything Going According to Plan|
|9-11||Breakthrough, +2 on next roll|
Roll 2d6 once at the end of the first week. If the search continues for a month or more, roll again at the end of every month. This does inject a slight possibility that the research will be protracted (reroll cost because of lost time,) but it's not as bad as the original table, where not spending ten times the base cost for research means that a player with bad luck may run out of funds long before any results are achieved.
You could, if you wished, apply the lost time result to the catastrophe result as well, but I figured a catastrophe is bad enough, by itself. The 1d6-2 roll means that a third of the time the catastrophe doesn't cause any damage. The GM has to be the judge on whether damage is more likely to be physical injury or fines and extra costs. Searching for information on the streets, for example, may get you a black eye if you ask the wrong questions, but it usually won't kill you, so it's better to just treat it as extra money spent to repair or replace furniture in a tavern or bribe the night watch to let you go.
A total success result, needless to say, should stop all further costs and rolls. You found what you were looking for.