I would look at a startup call Introhive. They have a unique approach to this issue based on personal connections.

*Disclosure - I do not work for them but I have friends who do.

Eoin's suggestion is a good one. If you make connection requests cold, be sure to include a detailed note, otherwise you'll be wasting your time. But as you said you're already using LinkedIn, I'm sure you know that.

IntroHive is good but it's really for enterprise customers who have an active CRM full of existing contacts.

A friend of mine has just launched http://www.thehonestcustomer.com It offers a unique approach to customer development in that it delivers surveys (pricing and customer development questions) to people who are your potential customers. You can define Industry, Company Size, and Job Title to target it very specifically. It's a good first step to creating the customer relationship.

If you want an eTro, just message me.

I use my linked in subscription and am always surprised by how easy it is to get a reply. It is a lot of work, but much easier than in the days of showing up in the lobby.

I would recommend sticking with LinkedIn.

Did you know that when using the Advanced Search on LinkedIn you can use a filter to only include people in Fortune 500 companies?

Set up a good profile page for everyone involved, make sure it has a photo.

Set up a company page, where you can also add a mini pitch.

Join Groups that contain your target market. You can send a message to group members.

Make connections, you need to expand your network to show up on their radar, as you can only see 1st, 2nd & 3rd
Join L.I.O.N & 500+ groups to speed this up

Don't Sell On The First Contact

How many times a lead is looking at your website and how often a lead is looking at your website are two great indicators of the extent and frequency of the customer pain. From there, it’s about connecting with the right person internally and showing them how your service or product can provide value». You can normally hone in on a specific person as your primary influencer or contact. To give you an idea, you may always reach out to the start-up’s head of sales.
Unlike working with SMBs, roles at Fortune 500 companies are not always consistent with those at other organizations. The vice president of marketing at one Fortune 500 could have a completely different role than her counterpart at another. Use LinkedIn, company bios, and personal websites to determine the best people to connect with based on responsibilities, KPIs, and relative authority. If you end up getting in contact with the wrong person, but they are open to a conversation, use them as a resource and advocate to get connected to the right person.
You can read more here: https://blog.hubspot.com/sales/5-keys-to-successfully-selling-to-fortune-500-companies
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath