I also have an 06 6-6 coupe. Mine went into 3rd gear fine most of the time but would pop out sometimes when I was giving her the beans. Sometimes it wouldn't go into 3rd when shifting at redline etc.. It's a pretty common problem that was easily solved by changing the trans fluid. Fresh OEM fluid usually works for people but thre is no guarantee if you have a bit of syncro wear. Using a friction modified synchromesh fluid is the answer.
Most of us from the forums use this: ACDelco 10-4014 Friction Modified Synchromesh Transmission Fluid (after I switched it fixed everything and never popped out or refused to go in at any temp or rpm again - it's been 6 years now and it still shifts perfectly)
Others have confirmed that Pennzoil synchromesh, RedLine Synchromesh, and Amsoil Synchromesh all work great too.
I also had a 6-6 and the 1-2 shift was fine in my car. Do you know about\have you put in the GM Syncromesh transmission fluid? 3rd gear was the biggest problem of my gearbox because of the defect in the gearset. Putting that fluid in made a huge difference. You look at the reviews and it's nothing but Honda and Acura guys bitching about their 1-2 and 2-3 shifts and how this stuff completely solved their problems.
https://www.amazon.com/ACDelco-10-4014-Friction-Synchromesh-Transmission/dp/B0026JK8FK
There's a known issue with the TL manual transmissions where it's hard to get into third gear. In many cases, though, substituting General Motors manual transmission fluid "fixes" it. Worked for me in two of my TLs.
Yeah I'm positive it even got to the point some acura dealers were changing out the fluid because the transmission was rebuilt and the problem persisted. A civic from the same era with a similar transmission also had a 3rd gear synchro issue and lots of them also switched the fluid which cleared the issue. You can find pages of people with the issue online going all the way back to 2004 when it was first released all the threads point to this being the fix and with my personal experience of having the same fluid for 100k miles with no hiccups I'd agree with them.
Heres the amazon link, the transmission takes 2.3 qts so you'll need 3 bottles https://www.amazon.com/dp/B0026JK8FK/ref=cm_sw_r_other_apa_glt_i_A1NNCM47BRC3TEKZGB47
I've seen changing the fluid help a lot, usually on older cars where the fluid is quite nasty.
I will echo what other people have said though - it sounds like the clutch might not be disengaging all the way.
My favorite fluid at this point is AC Delco Synchromesh Friction Modified. https://www.amazon.com/ACDelco-10-4014-Friction-Synchromesh-Transmission/dp/B0026JK8FK/
It was GM Synchromesh Friction Modified. There's also an AC Delco / GM Synchromesh without the Friction Modified Part. You don't want that one. The friction modifiers are what make it so good.
I'm not sure if you're in the US and what fluid you can get. I have other suggestions if you want.
Here's my recommendation:
Transmission Fluid
Since you are a fellow DIYer, follow this writeup: Acurazine How To
Use GM Syncromesh with Friction Modifier for MT. You can Google "GM Syncromesh Acura TL" and find lots of positive reports on using this fluid versus Honda fluid.
Here is the fluid I'm talking about: Amazon Link
Oil Leak
So potential leak sites on the engine include the oil pan flange to block, the oil pump (located between the crank pulley on the RH side of the car and the engine block), and the main seal between the engine and transmission. On my 140k '06 TL the oil pump was leaking, which coated the RH side of the engine, and the RH subframe with oil. Like this.
To fix this requires removal of the oil pump, which requires removal of the timing belt and oil pan. I needed to do the timing belt, so I bit the bullet and knocked it out. Here is a YouTube video of a pro doing this.
PS Pump
There was a bulletin from Acura to replace an oring on the pump. Part 13 on this diagram. The replacement oring is orange in color. There is an Acurazine post on rebuilding the PS pump as well, here.
GL with your UA6. I want to run mine to 300k miles.
GM Synchromesh Friction Modified (now called ACDelco) https://www.amazon.com/dp/B0026JK8FK/
This removed my 3rd gear grind in my ~150,000 Mile GSR
Lots of info about it on Honda-tech too.
Ac Delco synchromesh fluid I changed mine in my tl about 50k miles ago and haven't had a single issue since, mine was third gear.