So you finally got cast in something in or around New York. Awesome! Understand that if you're new to the City, this is a golden opportunity to show people your work. Even if you have an agent from your college showcase or a referral from someone, your agent needs to see you in action. Same with casting directors with whom you have a good working relationship.
Step one is to invest in some postcards with your headshot on them. DO NOT WASTE YOUR MONEY AT REPRODUCTIONS! Order them on the cheap at VistaPrint or a similar service. You're going to be sending your face through the mail. It'll be a miracle if you don't end up with a Hitler mustache on your lip by the time it arrives. And don't put it in an envelope. Many industry folk don't open unsolicited mail, and unfortunately, this qualifies. But if they see your face, they might say, "Oh, her! What's she up to?" And show up. Agents and casting directors go to theatre all the time.
If you have an email address for an industry person you'd like to invite, go ahead and use it. Attach your headshot and a link to the show and/or the production's promotional materials. However, be sure not to link to the ticket page. Check with management before you send out any material to see what their policy is with regards to industry comps. Most of the time theatres will offer them to industry, even if they don't offer comps to cast members themselves. But sometimes they don't. Generally, the more high profile the show, the less likely they are to offer free tickets. If this is their policy, you're going to have to cough up the ticket price yourself. (Only one! You don't have to pay for your agent's date.) However, this is a much smarter way to spend your money than, for example, a mailing, or a workshop, or an industry mixer. Presumably, they're seeing you in a role. It's an investment you should make.
Next, who to invite? This one's easy: everyone you can think of! Your agent (duh), and/or any agent you've met who told you to "keep in touch." This is exactly what they meant. Any casting director who has called you back and would recognize your face from the postcard.
Don't have any of those? The next investment you need to make is in a couple of classes or workshops at Actors Connection or One on One. In fact, this is the optimal time to be showing your face at those types of places. Think of it as your opportunity to present yourself as a working actor to people who could keep you working in the future. One of the unspoken rules of this industry is if somebody wants you, everyone wants you. Unfortunately, the reverse is also true. So take advantage of your good fortune. Target the CDs from theatres or TV shows where you want to work. Target the agents you've heard good things about. Yes, you can blindly mail all of these people a postcard, but they won't come. You could be crazy or awful. Best to make a personal connection first, mention your show, and then follow up with a postcard or email... or both!
This is how most people find representation, by the way. You need to be in something.
Accept that this line of action may result in some wasted expense. Not everyone you meet is going to come see your show. However, this industry is a game. To quote Hamilton, "When you got skin in the game, you stay in the game. But you don't get a win unless you play in the game." So take the chance. If nothing else, you can always write it off at tax time. But this is the sort of investment that could pay off big time.
One final thing: don't shy away from being a pain in the ass. You have to push yourself; you have to hustle. If you have an email address, follow up if you don't hear. Likewise, if you email and hear nothing, follow up with a postcard. Then maybe another email. And if industry people come to your show, follow up with them afterward. Send them a postcard thanking them.
A show in town can absolutely take your career to the next level. Use this opportunity wisely. Think hard about the type of career you want to have, and go full-out in that direction.