Talk to the GM about the action needed to designate prey
The rogue advanced talent hunter's surprise as its Benefit says
Once per day, a rogue with this talent can designate a single enemy she is adjacent to as her prey. Until the end of her next turn, she can add her sneak attack damage to all attacks made against her prey, even if she is not flanking it or it is not flat-footed.
Unfortunately, no action is listed for designating prey. Because it's an extraordinary ability, according to Actions in Combat designating prey takes a standard action to employ, but there is some debate. So far, the only answer seems to be to ask the GM.
However, the second part is clearer. The teamwork feat Swing About as its Benefit says
If you have a free hand, as a move action you can grab a willing adjacent ally who also has this feat and place her in any other space adjacent to you. This movement does not provoke attacks of opportunity. You can do this only with allies who are able to move freely (not grappled, held, prone, slowed, or otherwise suffering restricted movement) and who are of your size or smaller.
Note that, because of the way teamwork feats work, in addition to the party member doing the swinging, the rogue himself needs the teamwork feats Swing About and the feat Escape Route, the feat Swing About's prerequisite, but, if the rogue has all that, once per day (as per hunter's surprise) this could be done.
That's an awful lot of resources to devote to that one thing, though. And if you're going that far, be sure to get a weapon with the magic weapon special ability sneaky. (About which there's also some debate.)
Even if the rogue has the feat Improved Precise Shot, the rogue's foe can still have concealment. If the concealment's less than total, the rogue will ignore the miss chance, but the foe'll still have the concealment, so the rogue won't deal sneak attack damage. The feat Improved Precise Shot doesn't make the concealment dissipate; instead, the feat only mitigates its effects for the rogue.
Likewise, even if the rogue has the rogue talent sniper's eye, the rogue's foe can still have concealment, but the talent allows the rogue to deal sneak attack damage to the foe if that concealment's less than total. However, the rogue still suffers the miss chance from the foe's concealment.
Having both the feat and the talent means the rogue, when confronted with a foe with concealment less than total, ignores the miss chance from concealment and can deal the foe sneak attack damage.
Best Answer
As you observed, you can't trigger Underhanded in combat on its own
You are 100 percent correct that since they can only take a Standard or a Move action RAW (free actions are also allowed). At best the drawing of a hidden weapon can be made to be a move action (but you still only get 1 or the other) utilizing the Quick Draw Feat.
Bandit's can easily make use of underhanded
The rogue Bandit archetype gains Ambush(Ex) at the cost of Uncanny Dodge.
Thus the player would be able to (with Quick Draw) pull the weapon out with their move and still have a standard to attack the enemy with.
The Betrayer Feat might apply
However the wording seems to imply that combat begins after the betrayer attack, but then again Surprise is worded extremely vaguely (at your DM discretion).
Ostensibly the target of the Betrayer attack knew the other person was there, but not as a combatant and as a DM I would allow the attack to be treated as a Surprise round attack.
Possible workaround through items
My Pathfinder knowledge is insufficient to know if this is really RAW, but both the Sword Cane and the Switchblade can be drawn as a swift action. Whether this still applies when the weapon is hidden is something I'm not able to divine. Some people on Paizo's forums seem to think its workable and others don't.