$1,000/month is a poverty wage and will not make a meaningful difference in lowering the United States’s Gini coefficient. If people are serious as using UBI as a means of repairing our system of immense inequality and preparing for the oncoming automation apocalypse, then we need to get real about redistributing from the wealthiest corporations and persons to everyone via a more generous monthly disbursement ($2,500 would be in line with the income from a $15 minimum wage) fueled by a progressive tax. You simultaneously dampen the rapid accumulation of wealth by the hyper-rich and bring up the wealth of the poor. That is precisely what gets you a lower Gini coefficient.
Once you have this implemented, you then peg the monthly disbursement to economic growth, thereby preventing runaway income inequality from developing again. One of the biggest culprits in income inequality is that the rich soak up a disproportionate amount of the gains from economic growth, but if the UBI disbursement increases along with economic growth, that will be much less the case. Ultimately, after decades of economic growth, you end up with two classes of people: the rich and the very rich. That doesn’t sound so bad.