This is in response to Unemployment Benefits Decrease Poverty? Really?!
I'd propose an indirect mechanism, which can therefore explain how the correlation can fail to emerge. Unemployment insurance acts as a stabilizing force, allowing families to maintain their lives despite unemployment. Under normal circumstances this would be a temporary effect, tiding them over until they find a new job and begin paying back into the system. In this way, unemployment would have a slight impact on poverty by means of giving people money, but the larger effect would be on maintaining demand, thereby keeping the economy growing. It would therefore have the effect of smoothing out some problems so that free markets can reduce poverty.
This falls apart in a sustained recession. If there are too few jobs to find, then the insurance will no longer be stabilizing the economy. It will instead be propping up people's lives, without being of much benefit to economic growth. In summary, unemployment insurance is a brilliant idea if we're in a short downturn, but not in a long-term economic malaise. It may still be morally justified to help people, but it is not a recovery program.
Of course merely bashing a program isn't of much help. From that we'd conclude that the correct decision is to simply end it. That would crash the economy and ruin many lives. Or end them. It is always good to remember that people can starve to death or die of exposure. Lack of proper nutrition will have a life-long impact on a child.
What are the alternatives then? Or what are the supplements that can be used to improve the program?
First, it is always good to have people in the same place as jobs. Government could offer increased benefits to people who move to areas with lower unemployment or pay some moving expenses. The housing problems don't help, since they may make people feel or actually be stuck in their present locations. Increasing the mobility of labor would decrease unemployment. Economic improvements tend to compound. Lower unemployment will cost fewer resources at all levels of government, while also bringing in greater tax revenues. Even with the buffer of unemployment insurance, we can expect some rise in demand from a family with income than one without, boosting the economy further.
Eliminating rules such as the minimum wage, protections for unions, and workplace safety would make employees less expensive and therefore would encourage hiring. However, in a weak job market and without collective bargaining, there is no guarantee that workers will be paid enough to survive on.
Encouraging hiring would help as well. This could come on the demand side, so that employers see greater benefits from hiring. However, demand-side boosting requires a massive expenditure and we've not yet seen the political will for a sufficiently large stimulus. Alternatively, giving everyone a bunch of currency, using inflation to cause a redistribution of wealth from rich to poor, as opposed to the poor to rich movement we see with most monetary policy, would provoke a backlash from the politically-influential group known as "rich people".
Employers could receive tax benefits for hiring, such as from a pass on payroll taxes. However these tend to be deficit-increasing as the economic growth and resulting revenues don't offset the tax expenditure. The employment boost may end when the tax subsidy ends, and such subsidies tend to encourage hiring for low-wage jobs to game the system. If done at the state level, then it is merely part of the pirating of jobs with no actual economic benefit, an absurd practice known as "tax competition."
Rather than making hiring more profitable, government could use policy to make firing less profitable. A tax on layoffs would add an incentive to retain workers, though the retention would be proportional to the size of the tax, and would reduce business flexibility. It is good, after all, to be able to get rid of workers who don't produce as much as they are paid. Furthermore, if the tax were too high, then we could expect hiring to be reduced due to uncertainty about the need for labor.