This is a real tough nut to crack. A mission seems to be the right type of card to solve this problem, and the Design Team could enforce a cost of using a mission that is in and of itself not particularly helpful.
Here's the basic core of what I have in mind:
Event
To play this event, you must command one or fewer headquarters missions.
Plays in your core. Your personnel may attempt and complete any mission.
This event cannot be removed from play.
The most obvious problem is drawing this into the hand. Everyone succumbs to the occasional Rakal Shuffle, and Second Edition has been good about giving players the means to overcome that bad shuffle without becoming the exercise of "find card X" that eventually overcome First Edition. Keeping the cost at 0 would mitigate some of the "search cost" for this card, and there are means of grabbing events, but nothing completely bulletproof. (Nor should there ever be!)
Granted, I'm not as well-versed in all of the cards of the game as I should be. I'm very rusty, but it seems like this problem might need a solution from outside the realm of current mechanisms.
As Second Edition evolved in the first few sets, I recall there being some talk about special cards that might enter play before the game started. The concept would be something similar to the "seed cards" of First Edition. This might turn a few stomachs, but I think such a mechanism is warranted here. If we believe this kind of gameplay option is highly desirable, we should do what we can to make it easy on the player.
When I first heard of this seed concept, one word stuck to my brain and hasn't left since: Launch. It seemed like an appropriate kind of keyword to have around. I say "keyword" because (A) no way in h*ll do we need another card type ever and (B) it could be applied to any reasonable card type -- equipment, events, personnel, ships.
The first incarnation of Launch appeared on a few private dream cards. I could never get the syntax just right, but it was something like:
Launch: 3 (When you place your missions face up, you may play this card from your deck. Draw three fewer cards in your opening hand.)
As you can see, I set the cost to be a variable number of opening hand cards, which could be significant. Obviously, a Launch card would have to be something quite spectacular to warrant giving up that opening hand. Something extraordinary could have a Launch cost of 7, thus robbing a player of a turn.
I haven't really spent a lot of time thinking about it since then. But now because of this discussion I've let it marinate in the brain for a little while. I never really liked this, but I rarely like anything I concoct for this game. Why didn't I like this hypothetical concept? I can't say. I think I just didn't like meddling with the opening of the game, and the variable cost scale could allow two or three cards other than missions on the table before that player starts.
Was there something far more Second Edition-y I could do? Yes...
Launch (When you place your missons face up, you may play this card from your deck at cost 0. Remove one of your missions from the game.)
There is slaughtering of several sacred cows here. It seems quite ludicrous. Also, the syntax is off.
In light of this callout text, I think my earlier stipulation might be a bit too limiting from a gamesmanship perspective. Here goes nothing...
[1] *Seek Out New Life and New Civilizations
Event
Launch (When you place your missons face up, you may
play this card from your deck. Remove one of your missions from
the game.) Plays in your core. Your personnel may attempt
and complete any mission. This event cannot be removed from
play.
I removed the previous restriction on having only one headquarters because you are giving up a mission to use this. And, if a two-HQ player wants to try this, he should feel free to do so. Favor the Bold comes in handy there.
But, this is a mess and I'm proposing something completely off the table.
Enjoy.