After the party at the beach, you’ll get a new phone from DedSec. But first, you have to install some new stuff on it. Install all available apps from the App Shop since you should be able to afford them anyway. Install the Nudle Maps first then open your map. Look for a clothing shop icon and fast-travel there. You can fast-travel to other establishments like kiosks, restaurants, and shops. The nearest one with the good selection of outfits will be the Torque Rat to the northeast. If you have DLC outfits and the like, you can access them from any clothing store or the Hackerspace locker room.
Head to the shop and start changing your look. You really need to buy at least one pair of pants for the objective to be completed; equipping DLC outfits (like the White Hat pants) won’t count. Just get any pants then change it to something else if you like. And just like that, the mission is now complete.