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Experience buying Treasury Bills

Could you share your experience buying T-bills through Fidelity in the secondary market? There are a lot of data points and I want to understand what I should be looking out for.  Looking to purchase $50K for 6 months. 
is it better to purchase at the Treasury auction vs. secondary market. I wonder if it will make a big difference at these dollar amounts, given the efficiency of the bond market.
- My observation is that based on depth of book, generally it seems that the more you purchase the better is the interest rate.
- in the table below market price for a 25 lot bond is 99.570 but one can use limit price to try for a higher interest rate.
- Spread to Treasuries: given that this is 1/100th of a percentage point, this is not material at the $50K dollar amount?
Anything else I should look out for?
Thanks in advance.
You asked about Treasury bills but posted the book for a (now) 5+ month Treasury coupon bond.  6mo bills are typically far more liquid, and most retail investors buy them free in the weekly auctions.  FWIW, one basis point on a six month bill is $50/million ---- on a $50,000 face purchase that 0.01% is worth $2.50.
I confess I call all the treasury offerings T-bills. I am typically looking for something maturing soon on the secondary market, so I don't care how long the original maturity was. I have bought (and sold) a few treasury offerings on the secondary market. I especially appreciate that the market is very liquid, so you can always unload your bills/ bonds/ notes quickly.

In the original example you cite, the highest bid (for the 25 bonds) is 99.507 and the low ask is 99.570. If I was interested in this issue, I would probably initially bid 99.55. I have had pretty good luck shaving a bit off the low ask. The default order is fill or kill, so to me, you submit a lowish offer, it gets bounced in twenty minutes, you have another data point, and you can try again. Based on my limited experience, I would guess in this case, you could "fairly confidently" get it at 99.56. As other posters correctly point out, the difference in total cost is pretty insignificant. And if you don't get the response you want, you can always try another maturity date.

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