Without knowing your exact use case, I can only assume you're not just searching for one record. Name is not a contact field in Clio and is probably a combination of any/all of first_name, middle_name, last_name, title, prefix. If the initials returned in your example is only "J", but name returns "James Town", this should provide some insight that "name" is not a reliable method to query on. https://app.clio.com/api/v4/contacts.json?fields=id,first_name,last_name&limit=200

If the limit is your issue, use the ['meta']['paging']['next'] provided in the response json to loop through all of the pages. Do the filtering application side, and then if a match is found, use the id to retrieve the full record: https://app.clio.com/api/v4/contacts/{id}.json. This would be more accurate and consume significantly less resources on the server.

If you're trying to do any sort of autocomplete or smart filtering, get ready to get banned from the API without carefully considering your approach.