It's easy to underestimate this card. Gerrark made a nice review above, but I have a few extra things to point out.
For one, from the unofficial FAQ we know that you resolve all effects on the first card before even choosing the second card. As an example, if you have 6 creds, one click, a tagged Runner plus this, a Hedge Fund and a Restructure in hand, you could play the Subcontract, first play the Hedge Fund then choose to play Restructure, now that you can afford it.
That may seem underwhelming, but where it gets wild is the synergy with Election Day. Yep, going by the same logic as above - you can first Subcontract, choose Election Day (assuming you have at least one other card to trash) then play one of the cards you drew with Election Day. Theoretically, this means that it is possible to, in a single click, cycle through 15 cards and then get to play one of them (Subcontract -> Election Day + Subcontract -> Election Day + Subcontract -> Election Day + one chosen card). Of course, that is incredibly unlikely, but it is absolutely realistic to use this to reach for additional damage that you need to win a game right now, and depending on how your deck is built, the chances of seeing some damage in five cards can be quite high. At worst, it's a good way to get to play some of your hand before you Election Day the rest away.
Another great synergy is with Consulting Visit, an absolute staple of any Weyland-oriented kill deck in my opinion (and influence-free to most anyone with dark green tendencies!). Assuming an already-tagged Runner (Midseason Replacements or Posted Bounty before your first click), you can play Subcontract into double Consulting Visit (costing one click for the Subcontract and then one click for each Consulting Visit) - keep in mind that normally double Consulting Visit is impossible due to the click cost, but can be very powerful, e.g. to fetch double Scorched Earth or a double Punitive Counterstrike. Alternatively, you can spend your first click to tag the Runner, then Subcontract to play Consulting Visit plus a kill card like Scorched Earth or Traffic Accident, which is generally a kill on anyone.
With Plascrete Carapace heading for the dumpster, a surprisingly powerful side of Subcontract is that it does not allow any reaction that is not a prevent effect. Most notably, if you Subcontract two Scorched Earth or a Scorch and a Traffic Accident, the Runner will not have an opportunity to use their Sports Hopper to save themselves. Depending on what new damage prevention cards appear in the next few packs, this might be a huge deal.
Finally, this card hands-down takes the cake for "Most damage possible in a single click", with hilarious combos like 3x Subcontract into, say, three Punitive Counterstrike and a Scorched Earth. Mostly for hilarity, but as illustrated above, being able to pump out a huge amount of damage in a single click might become very valuable going forwards.
As a side note, there's a slight edge case with the upcoming Boom!. Assuming the Runner starts with one tag (unable to clear one or Posted Bounty before first click), you can use Subcontract into SEA Source then a Consulting Visit for your Boom!. Alternatively, if the Runner starts with no tags you can SEA Source, then Subcontract for your second SEA Source and your Boom!. While it certainly is an edge case, it might just become more relevant with Midseason Replacements gone.
The way I tend to play this card is to treat is as disposable unless I am sure I can get value from it, and I have no problems ditching it to Archives or using it to just get an extra click to play some econ cards unless I know I will need it for kills. However, it has honestly been performing much better than expected and is consistently enabling one-turn kills or fuelling double-Election Day-into-the-burn-you-needed-to-finish-this-game-right-the-heck-now miracles, and while I would never recommend keeping it in your opening hand, I have actually started prioritizing this card even over some kill pieces in certain situations.