It takes two to tango. The only way you jump ahead of the line is to sell at a discount.
Not really. What you do is you survey the field, find the team that is most desperate from the jump, see if you can push them up a little, and set the market.
The Flames did not sell Lindholm at a discount, even though he was the first big name C off the board (real good player in a thin market, but they still got surplus value). Last season, the first big defender to move was Orlov (+ our friend Hathaway). The Capitals got a terrific return for him 2 weeks before the deadline—1st, 2nd, 3rd and a cap dump.
Hell, the Flyers were the first team to make a big move last summer, dealing Provorov. And they got great value there, too! In-season, though, the player has slightly more value to the acquiring team if they're acquiring him earlier, because they will get more games out of the player prior to the playoffs. But that value will become negligible in a week or so.
There's this thing about concepts like "discount." They're always relative to what's happening in the market. The team that sets the market usually is not selling at a discount, because there is no frame of reference. Likewise, the acquiring GM has no comparables to point to, and can be asked a higher price while all of the potential bidders are still in the market.
These players aren't worth the same thing to different teams. There's no Kelley Blue Book Value on 2nd pairing defensemen. There might be 6 teams that are interested in your guy: a couple of them are REALLY interested and the rest are just willing to kick tires as a backup plan. If the Leafs and the Bruins, say, are the two teams pushing for your player the hardest, and suddenly the Bruins go get some other guy, then the Leafs are just competing against the 4 other teams that see your guy as an alright backup plan and then the price is suddenly way, way lower.
This has been a trend for some time now, actually. How many years in a row have people complained that the deadline itself was a dud and a waste of time? We've reached the point where very few teams are still active on the day itself, because the trend has shifted earlier. The "desperate GMs pay more!!!!" thing is a myth. There are very few Fletcher-level fools in the offices these days.