In addition to doing POST and GET requests, I have had sucess doing PUT and other request methods by modifying the header using the "X-HTTP-Method-Override" method. More info on HTTP headers here https://en.wikipedia.org/wiki/List_of_HTTP_header_fields
For example, to do a PUT request:
Within JSON Tool,
Call Type: POST
POST Method: Custom JSON Payload
...enter your json payload
under "Request Header" section:
Name : X-HTTP-Method-Override
Value: PUT
*Try substituting "Value" with DELETE, OPTIONS, TRACE, etc. for other methods. (I've used PUT & DELETE sucessfully, but it's probably dependant on the API you are communicating with)