On several occasions our heroes in Stargate Atlantis worry that their gateship may have been detected between exiting the stargate and activating the cloak.
Why don't they just cloak before they go through the gate?
We see in various episodes, such as "The Siege, Part 3" that the use of a cloak negates any shielding the vessel may possess. Given that they're entering an unknown situation (where the enemy may simply be firing continuously at the gate), it makes tactical sense to use the main shields when entering the gate, then cloak afterwards.
In the episode "The Shrine" we see precisely this situation. I've edited for brevity...
In the Gateroom, a M.A.L.P. trundles into the open Stargate. As it reaches the other side, Richard looks at the video footage which it is sending back. Two Wraith Hive ships are on the ground on hillsides either side of the Gate. He reports on what he's seeing.
WOOLSEY: The M.A.L.P. is clear of the Gate and they're firing on it from both sides of what appears to be a deep valley.
The Jumper lowers down into the Gateroom.
WOOLSEY: It's deploying smoke. You should have complete coverage of the area in five seconds.
(The Jumper flies into the Gate.)
TALUS. The Jumper exits the Gate and soars forward through a thick cloud of smoke. Its cloak can be heard deploying as it races up into the sky. It turns around and heads back the way it came to take a look at the two Hive ships.
TEYLA: Do you think they detected us before we were able to cloak?
SHEPPARD: No, I think the smokescreen worked. Hopefully they think we just sent the M.A.L.P. through ahead of us and changed our mind when they blew it up.
In several scenes we can see writing in Ancient across the top of the jumper console. Two images are here, taken from the Season 1 episode, The Defiant One *, show the left and center of the console, respectively,
Rather than being actual Ancient, the writing is just English with Ancient characters. Piecing together the text from various images of the console, we obtain the full ancient text (as it would appear to someone sitting in the jumper reading from left to right):
Fortunately, there is an Ancient to English alphabet guide on Wikipedia:
The translation is as follows**
Gate protocols must be set before launch sequence.
Never attempt gate reentry when in cloak m??? [mode?]
So, in short, they don't cloak when going through the gate because the Ancients wrote clear instructions in the jumpers not to.
Of course, one might wonder why the Ancients wrote instructions on the console in English using the Ancient alphabet, but that's another question.
* I haven't been able to get a good shot of the right side of the console. I did all of the translations off of images in The Defiant One, but there is no moment when the right side of the console is visible... I freeze framed and went step by step to get each character. If someone can find an image from another episode that shows the right side clearly, please do upload it.
** I've inserted spaces and some capitalization, which the Ancients didn't seem to use.
I don't think they ever explicitly say, but because they never go through the stargate while cloaked I don't think they can. My guess is because Ancient cloaks are extremely powerful, hiding puddle jumpers both from vision and also sensors. In fact, the only time a cloaked jumper is sensed is by a wraith ship using a ZPM -- and even so the wraith ship didn't have a perfect idea of the puddle jumper's positioning because it missed the shot it took at it.
But because it hides them from sensors-- the stargate might not realize they've gone through, so it can't store them in it's hyperspatial buffer. The Ancients probably had the technology to sense cloaked jumpers (so it would work with the stargate) but it takes so much energy to sense a cloaked jumper and get even a vague position that it was easier to just say "Don't go through the gate while cloaked."
To rule a few things out:
1.) It's not "just part of the automatic launch sequence". In Season 1, Episode 17 "Letters from Pegasus"
[13:50] Sheppard: I'll take us out of stealth mode for the pass through, dial the gate.
In that scene they are returning to Atlantis. One popular theory is that it's not cloaked because of the automated systems that take over when a puddle jumper approaches the stargate -- but we can see in this episode he is doing this manually. There are automated systems for getting through a stargate but it doesn't seem to include uncloaking.
Per Season 2 Episode 10: "The Lost Boys"
[44:40] Sheppard: The Dart went on autopilot the second I got close to the hive. ... That's what Jumpers do when they gate back to Atlantis.
2.) It's not because the shield/cloak makes the jumper too big, or that shielded items can't go through.
Season 4, Episode 17 "Midway" @22:05 The wraith send through a shielded weapon.
Season 3, Episode 11 "The Return: Part 2" @3:08 They fly the puddle
jumper through when it is completely shielded. Of interesting note,
it doesn't look like it's shielded before they enter the stargate, so they must do it at some point while in the wormhole because as they exit they are already shielded.(Which doesn't make sense, and is probably a mistake.)