Looks like you can use the method refresh_request( ) on the CL_HTTP_CLIENT instance to reset your request object. I found that calling this method allowed me to send multiple requests on the same open HTTP client.
Here's how it works:
* Retrieving response from first request
cv_doc_info = lo_http_client->response->get_cdata().
* Reset request for this client
lo_http_client->refresh_request().
* Setup the next request
lo_http_client->request->set_method('GET').
cl_http_utility=>set_request_uri(
request = lo_http_client->request
uri = lv_new_uri
).
* Submit request next request
lo_http_client->send().