Just to be clear on my thoughts, I think Commander Thompson ordered smoke making a little too early. My thoughts are based on some facts, some experience, and of course the speculation that always creeps in.
At 0614 (local time) Itasca receives and logs the following message from Earhart: "WANTS BEARING ON 3105 KCS // ON HOUR // WILL WHISTLE IN MIC ABOUT 200 MILES OUT// APPX// WHISTLING." I imagine her estimated range from Howland is based on the DR line that Fred Noonan is keeping, based on their last fix during the night. (Which we don't know or have anyway of knowing.)
The deck log of the Itasca indicates "Vessel began laying down heavy smoke to assist Miss Earhart." This entry appears immediately after the 0614 radio message, and the similar information regarding the 200 mile distance appears in the deck log with a local time of 0614.
Commander Thompson orders the fireroom to produce black smoke. This was most likely done by increasing fuel supply of bunker oil to the burner fronts, and throttling back the air registers to the boiler. (The whereabouts and contents of the engineroom logs are not known to me right now.) There is probably a delay of only a couple of minutes before the fireroom begins making smoke. The local wind on the surface is strong, so the term "laying down" smoke is pretty accurate. The local wind is flattening the smoke and dispersing it as it drifts away from the ship, streaming it into a horizontal smudgy haze and eventually dissipating altogether.
The fireroom makes smoke for about 30 minutes. At some point, the senior watch stander in the engineroom is getting concerned about the safe operation of the boiler with so much bunker oil being burned in a black smoke condition. He probably called the bridge and requested permission to stop. The commanding officer or Officer of the Deck would either grant it or refuse. If ceasing smoke was refused, at some point the Chief Engineer is going to want to talk to Commander Thompson and point out the dangers by continued smoke making, and Commander Thompson has to make a decision between helping Amelia Earhart find Howland Island, and endangering the Itasca.
Meanwhile, the radio room is struggling to get contact with the Electra. The ship is packed with reporters and various civilian guests, and the watchstanders on the bridge of the Itasca are concerned with topside operations such as monitoring Itasca's position westward of Howland Island, and the boats that were launched carrying the landing parties to Howland. It's not surprising that the deck log omits any reference of when the smoke making stopped.
I don't know whether a compromise was reached with Commander Thompson as to smoke making. Perhaps he ordered it intermittently, or perhaps it ceased altogether. The engineering logs would help.
At 0645, around the time the Engineering Department would want to stop making so much black smoke, the Itasca radio personnel log the following: "PSE (please) TAKE BEARING ON US AND REPORT IN HALF HOUR -- I WILL MAKE NOISE IN MIC -- ABT 100 MILES OUT."
Again, I think this estimate of range from Howland is based on the DR. (I do find it hard to believe that the Electra has covered 100 miles in half an hour. This reflects that Amelia is bad at putting times on her fixes and estimated positions.)
James Kamakaiwi later reports that the Itasca was making huge clouds of smoke. That tells us that smoke was being made, but not when or how long. A reporter later notes that when the Itasca leaves Howland to search, she is making smoke. But that would be typical, because boilers in those days did not have the automatic controls that we have today. As the ship began ordering speed, the Itasca would drain current on the electric motor powering the ship's screw. This would increase steam demand on the turbine generator, which would in turn increase demand on the boiler. The fireroom personnel must then increase feedwater flow to the boiler, and increase fuel and air to the burners. The ship would make smoke during acceleration, until the fireroom was able to get the water, air, and fuel flowing at a rate to meet the demand. Then the smoke would taper off.
At 0742, Radioman Galten's radio log has the following: "KHAQQ CLNG ITASCA WE MUST BE ON YOU BUT CANNOT SEE U BUT GAS IS RUNNING LOW BEEN UNABLE TO REACH YOU BY RADIO WE ARE FLYING AT A 1000 FEET." Of course, we don't know when Amelia Earhart descended to 1000 feet, or what her altitude was at the 0614 transmission of her being 200 miles out.
This is almost an hour and a half after the deck log indicates smoke making began. From all of that, my conclusion is that Commander Thompson started smoke making early. He should have started at 0730, not 0614, but I can understand his decision given the amount of confusion over when Amelia left Lae, where she was, her confusing estimates of her distance from the island, and logical assumptions (which were probably erroneous) that the Electra was travelling faster than it actually was.