If your target audience uses LinkedIn regularly, it's time to use InMails.
LinkedIn paid accounts offer their users a limited number of direct messages to other LinkedIn members (even if you're not connected with them).
Even better, these InMails guarantee a response - and if you don't get a response, your credit is refunded to try again.
To get the most out of this service, try to create a short and pleasant InMail template.
To stand out in an overloaded inbox, be concise, honest, and direct.
Don't go straight to a sales pitch, but instead try to offer an advantage to your prospect: a free consultation, a white paper, or personalized advice.