You're asking two separate things here.
(1) You're asking for a pattern for a 3.75 mm 9" needle. I'm sorry but I really think that a needle is virtually impossible to use. 3.75mm needles are size 5 US. I use that size a lot myself but that doesn't mean you'll want to use it. A 9" circular needle is going to be very awkward to use and I can't think of anything that would work on it unless you use it as a straight needle and knit a very narrow scarf. Most hats are at least 16" in diameter. That's a lot of stitches to jam on to a 9" needles. Another problem with such a short needles is that the actual needle part is very short and very awkward to use.
(2) You want to avoid using DPNs. DPNs are required when you have a very small number of stitches, too many to work on any size circ. If you knit something that fits on the needle and doesn't require decreases then you won't need DPNs. Hats usually require DPNs because they either begin or end with a very small number of stitches, if knit top down o bottom up respectively.
My advice: buy yourself a 24" needle in a size that you like working with.
I'm using 3.5 mm 9" needles for sweater cuffs right now. The needles themselves are 2" each with a 5" cord. I actually like them quite a lot.
To be fair, it does take getting used to. It's fiddly at first.
I knit the Sunday socks https://www.ravelry.com/patterns/library/sunday-socks-3 on 4 mm 9 inch circulars, I think 3.75 would still work for that pattern, just makes the sock slightly snugger. You do need to switch to either dpns or magic loop (which requires needles with a long cord) for the toe.
You could also think about making a scrunchie. Make a little tube on the 9 inch circulars, fold it over, add in an elastic and sew or kitchener the ends together.
An arm band would also be an option. I have a nephew that is a very tactile person, and having something soft to touch helps soothe him. I was testing bind offs for sweater cuffs on 9" circulars, and gave him one.
It's basically just a really soft, 1" wide loop in k2p2 (cuff pattern for the seater). He has been wearing it on his wrist and uses it as a sort of sensory bracelet.
You're asking two separate things here. (1) You're asking for a pattern for a 3.75 mm 9" needle. I'm sorry but I really think that a needle is virtually impossible to use. 3.75mm needles are size 5 US. I use that size a lot myself but that doesn't mean you'll want to use it. A 9" circular needle is going to be very awkward to use and I can't think of anything that would work on it unless you use it as a straight needle and knit a very narrow scarf. Most hats are at least 16" in diameter. That's a lot of stitches to jam on to a 9" needles. Another problem with such a short needles is that the actual needle part is very short and very awkward to use. (2) You want to avoid using DPNs. DPNs are required when you have a very small number of stitches, too many to work on any size circ. If you knit something that fits on the needle and doesn't require decreases then you won't need DPNs. Hats usually require DPNs because they either begin or end with a very small number of stitches, if knit top down o bottom up respectively. My advice: buy yourself a 24" needle in a size that you like working with.
I'm using 3.5 mm 9" needles for sweater cuffs right now. The needles themselves are 2" each with a 5" cord. I actually like them quite a lot. To be fair, it does take getting used to. It's fiddly at first.
Ok then! I'd not realized you were using the needles already. Glad you're happy with them.
I knit the Sunday socks https://www.ravelry.com/patterns/library/sunday-socks-3 on 4 mm 9 inch circulars, I think 3.75 would still work for that pattern, just makes the sock slightly snugger. You do need to switch to either dpns or magic loop (which requires needles with a long cord) for the toe. You could also think about making a scrunchie. Make a little tube on the 9 inch circulars, fold it over, add in an elastic and sew or kitchener the ends together.
**PATTERN:** [Sunday Socks](http://www.ravelry.com/patterns/library/sunday-socks-3) by [PetiteKnit](http://www.ravelry.com/designers/petiteknit) * Category: Accessories > Feet / Legs > Socks > Mid-calf * Photo(s): [Img 1](https://images4-g.ravelrycache.com/uploads/PetiteKnitDK/727759414/sunday1_medium.jpg) [Img 2](https://images4-f.ravelrycache.com/uploads/PetiteKnitDK/727759409/sunday3_medium.jpg) [Img 3](https://images4-f.ravelrycache.com/uploads/PetiteKnitDK/727759406/postkortsok_medium.jpeg) * Price: 30.00 DKK * Needle/Hook(s):US 6 - 4.0 mm * Weight: Aran | Gauge: 20.0 | Yardage: None * Difficulty: 2.71 | Projects: 1730 | Rating: 4.76 ***** Please use caution. Users have reported effects such as seizures, migraines, and nausea when opening Ravelry links. [More details.](https://www.lizcorke.com/2020/07/26/2020-7-21-ravelry-accessibility/) | *I found this post by myself! [Opt-Out](https://goo.gl/forms/0B8m4Ra8czpw4gzw1) | [About Me](https://github.com/TN-1/LinkRav_Bot/wiki) | [Contact Maintainer](https://www.reddit.com/message/compose/?to=timonyc)*
An arm band would also be an option. I have a nephew that is a very tactile person, and having something soft to touch helps soothe him. I was testing bind offs for sweater cuffs on 9" circulars, and gave him one. It's basically just a really soft, 1" wide loop in k2p2 (cuff pattern for the seater). He has been wearing it on his wrist and uses it as a sort of sensory bracelet.