There are a large number of games set in this general genre, thanks to the popularity of X-Files and similar media set in that genre in the 1990s. I own a lot of them, thanks to a decade-long game buying and playing binge with my first roommate out of college!
The first batch we should look at are the closest to Pathfinder in nature. Alternity was a far-future sci-fi RPG that TSR published, and they put out a modern day version called DarkMatter in 1999. Like most of these games based on popularity of the X-Files, it included rules for aliens but also psychics, magic, and other supernatural phenomena. Alternity was cancelled by Wizards of the Coast, but they published both "d20 Modern" and "d20 Future" versions of the new 3.x D&D d20 - and then they released a d20 Modern version of DarkMatter in 2006! It is shorter than the original and thus has less content, though, so if I were to run this again I would likely just run the Alternity version (and there's a lot of fan support at alternityrpg.net), but either are workable. There are actual attempts at a more Pathfinderized version of d20 Modern at d20modernpf.com (they also have a product) and pfmodern.wikia.com if you want the game to be really, really, super-Pathfindery. You can use these for "combat and underground bases" play but they're not 100% about that , they're designed to support that and other kinds of play in the general X-Files space.
Though this set of games is very d20-ey, and supports heavy combat play acceptably, they have a strong supernatural element to them and didn't get much adventure support. I don't know that pdfs of any of these are purchasable, but I see books from these lines in Half Price Books and such still nowadays.
Another set of games that is not d20 based, but is otherwise rules-crunchy and strongly supports the style of gameplay you are looking for, starts with Conspiracy X, another X-Files type game, in 1996. This has a very strong conspiracy aspect to it and "covering up the truth" is a major theme. And big guns. It includes both aliens and the occult, and had its own system. At about the same time, Steve Jackson published GURPS Black Ops in 1997. Its taglines are "Find The Truth - And Kill It" and "The Greys. Vampires. Evil Psis. Better Bring A Spare Clip." GURPS is a well known universal system and the game is very much "hunt aliens and monsters" focused, with a slight tongue in cheek (not as comedic as Men in Black). The respective publishers must have seen the chocolate and peanut butter nature of these two properties because they came together into a full-sized (285 pages) GURPS Conspiracy X book in 2002, which has a solid system and detailed support for fighting the Grey conspiracy. There is a newer version called Conspiracy X 2.0 in 2006 but all I have on that one is a Free RPG Day quickstart product so I can't really speak to it. This newer version uses the Unisystem, a system Eden Studios uses for most of its games. These are closer to X-Com in style, though you have to steer away from the supernatural parts - but You can get PDFs of all these from drivethrurpg.com.
Other games begin to diverge from the main thread a little, but have some strengths depending on what you're willing to work around.
Delta Green is a modern version of Call of Cthulhu that focuses on special ops and government conspiracies surrounding the 'alien' threat, where the aliens are those of the Lovecraft Mythos (Mi-Go, etc.). Whether this feels more like aliens or more like supernatural horror is a bit up to the individual. Use the huge "Delta Green: Countdown" adventure to really get that race-against-the-apocalypse experience!
Dark Conspiracy is a similar conspiracy/combat game but it's very much more about the supernatural and monsters and such. It has aliens too, but it's mostly about the modern world becoming "the Demonground." You can get the products in pdf from drivethrurpg.
There was an actual licensed Men In Black RPG (you know the Will Smith movie) that used the d6 system - I don't get the impression you're looking for that level of camp however.
Also, there are a variety of far future "fight the aliens in open combat" military games - Aliens, Bughunters, Starship Troopers - but that's farther afield from the X-Com experience. I mean, pretty much any full sci-fi game you can choose to hunt some aliens in, but they lack the "sub rosa" nature of these other games that are more purpose built for it.
One possible modern alternative (and a game that's actually in print, unlike all the others) is Night's Black Agents, a GUMSHOE system game that is intended more to be action/superspy/conspiracy/vampire killing - but they take a very general view of what the "vampires" are and some of the point is for you to pick something weird and random for the players to unravel in a given campaign; they can be time traveling aliens or whatever. This would provide more of a modern storygame take on it - they have great support for engaging with a conspiracy (cribbed from Blowback, the game inspired by Burn Notice).
You may also want to look out there at fan provided hacks - a lot of people have been interested in X-Com RPGs over time, and though I don't know of any complete games, there have been some pretty extensive explorations, like this Cortex Plus one. http://exploring-infinity.com/x-com-rpg/