It turns out automatically reinvesting Treasury Bills upon maturity is pretty straightforward, according to this TreasuryDirect link. I must say, there is a lot of information on all those government sites, but they sure make it hard to find it!
For example, for 4-week T-Bills, they both mature and issue on Thursdays. But if you set your maturing T-Bills to pay out into a Certificate of Indebtedness (C of I), and your to-be-issued T-Bill to fund from the same C of I, then the maturing T-Bill will first pay out money into the account before the 2nd one takes it out. So (most of) your money will effectively be “reinvested” into another T-Bill.
I say ‘most of’ because it seems like there will be some money left over. Since T-Bills are sold discounted and in increments of $1,000, you may pay $997 for a T-Bill worth $1000 at maturity. Then the next T-bill will also cost about $997 dollars, leaving you with a few bucks left over (your interest). So the interest doesn’t reinvest, since you can’t buy a $1003 T-Bill. Bummer. But you can just sweep that money into your Emigrant account or whatever.
What’s a Certificate of Indebtedness? It’s something that take 5 tries to type correctly (try it!). It’s also a fancy word for a non-interest-bearing account where you can hold funds to be used later to buy Treasury securities. It would be sweet if they paid some sort of interest on it, but no dice.