Maybe, but it needs a particular setup, buy-in from the GM, some suspension of disbelief, a willingness to outright abuse the rules, and a, um, flexible definition of when time passes during a combat round
Essentially, we're going to abuse the mechanics here in a way that doesn't make sense if you try to explain it outside of game mechanics. In particular, we're going to abuse the timing of the combat round.
So, you can't just say that you're casting Planar Binding at the very moment that the conjuration is finished, because that will almost certainly have just the slightest bit of delay (and thus creating that "one second short" situation you mentioned). Unless of course your GM just says "yeah, that works", in which case you don't need this. At the very least, I know that I wouldn't give that to you.
So instead, arrange to be in combat as the conjuration finishes (how you do that is up to you), and make sure you can stay in combat for a full hour. In your first turn immediately after the conjuration is finished, you begin casting Planar Binding.
Then, you need to stay in combat for 600 turns and not break concentration, and now your spell, somehow, finishes just before the summoning wears off!
But how does this work?
Well, a battle round represents six seconds of simultaneous combat. However, we're abusing that simultaneous part with the use of turn order. Even though everything in a battle round technically happens at the same time, everything does end up having a particular order mechanically.
So, to pull from your example, your druid finishes the conjuration and two woodland creatures appear. Your wizard, during their immediate next turn, begins casting this spell.
At this point, you're now in the weird physics- and time-breaking situation in which your wizard has already spent six seconds casting Planar Binding on a creature that the wizard summoned zero seconds ago! From here on out, your wizard, at least from a purely mechanical perspective, gets to experience those six seconds just before your druid does! And since your druid's spell wears off one hour after the end of the druid's turn, that means it wears off only at the druid's 600th turn!
Incidentally, Planar Binding finishes casting during your 600th turn. And, wouldn't you know, you just happened to get that one turn head start, so now your 600th turn will happen just before the druid's 600th turn! And thus, you will finish Planar Binding just before the conjuration wears off!
Of course, this does mean you have to stay in combat for 600 turns. I don't know how you'll manage that, but this is very important. The moment you fall out of combat, combat rules stop applying, and common sense makes this once again impossible to do. This will not be easy to do.
Results may vary
This method of course does depend heavily on what your GM is willing to tolerate, and what kind of game they want. A GM who likes more gritty or realistic games would probably not allow this, since it actually makes zero sense outside of counting combat rounds. These GMs will follow the idea that since you started casting Planar Binding after the conjuration was finished, then it's impossible to finish before it wears off regardless of how the rules can be twisted around.
On the other hand, a GM who likes following rules to the letter, with little or no regard to those rules' implications regarding the game world's physics, would probably allow this interpretation - you just have to convince them that this is indeed how the rules work.
There's also the chance that you might impress your GM with the sheer audacity of this plan, enough for them to say "Sure, if you can pull it off and stay in combat for 600 turns, I'll give it to you."
Why not just do this outside of combat?
Because Planar Binding requires a target, and the target you have in mind doesn't appear until after the conjuration spell is completed. Thus, you cannot begin casting until after the conjuration's duration has begun.
About the weirdness of the passage of time, and the assumptions I made
This answer relies on an assumption that the six second time unit only applies during combat, and that it cannot be taken advantage of unless combat rules are active. In other words, I assume that common sense rulings are in play if the rules give no indication otherwise.
If we assume instead that the six second unit of time can apply outside of combat, then this answer becomes unnecessarily complicated, and all one has to do is say "I begin casting the moment the spell is finished," and then maintain concentration.