Unfortunately, in May 2015 they removed that option for people with a free Basic account. Now you have to pay for one of the Premium accounts to be able to use InMails (you get a different number of them every month depending on your subscription level).
You can purchase a premium plan on a monthly basis and they do have a 30-day free trial option. As a premium subscriber, you can also still purchase up to 10 additional individual InMails beyond your normal allotment for $10 each. So this may be an option for you if you desperately need to contact people and can send them all within the same month. Also, they will re-credit you with an InMail if you receive a response from the person you contacted so you could end up with more. However, I’m not sure what happens if the person responds to you after your subscription is up.
LinkedIn also used to let you send unlimited messages for free to any members of groups you belong to. In June 2015, however, they announced that you can only send up to 15 1-to-1 messages to fellow group members per month, and that limit applies to all your groups put together, not each one.
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