Historical Descriptions
This is a summary of the historical source information related to the Hunley's torpedo.  It's not exhaustive but captures the pertinent information.  The quoted text is from The H.L. Hunley in Historical Context web page unless otherwise noted.   

     The original Hunley concept was to tow the torpedo on a long line, dive under the target ship and draw the charge against the target's hull.  This approach was abandoned when the torpedo drifted "too near the wrong boat" as Alexander recounted, perhaps euphemistically, in 1902.  In the same memoir he describes the new approach this way: "We then rigged a yellow pine boom, 22 feet long and tapering; this was attached to the bow, banded and guyed in each side. A socket on the torpedo secured it to the boom."  In 1925, Corporal Daniel McClaurin provided a slightly different description: "As I recall, the torpedo was fastened to the end of an iron pipe, about two inches in diameter and twenty to twenty-five feet in length, which could be extended in front and withdrawn at ease by guides in the center of the boat to hold it in place."  The apparent discrepancy between the two descriptions is partly alleviated when Alexander's drawing, published with his memoir, is examined.  It notes the "butt end of torpedo boom" attached to the top of the bow, drawn with a stylized wood grain, and a smaller, un-referenced object attached to the bottom of the bow and angled up (see image #53544 on the Submarine H.L. Hunley images page).  The many years passing between the 1864 events and these reminiscences makes some details suspect, but I believe the basic descriptions can be trusted. 
     Accounts differ as to how the torpedo was to be detonated.  Describing the towed torpedo system, Alexander mentions detonators on the torpedo itself, "one of the triggers to touch the vessel and explode the torpedo", a logical approach.  Also describing the towed torpedo, George Gift writes in a letter to his fiancée "The steersman has a string by which he can explode the torpedo by giving it a jerk".  Several accounts that indicate concern the submarine would be drawn into its target when the bow-mounted torpedo exploded imply impact triggers like Alexander's detonators.  This was the approach used for other Confederate torpedo boats.  In 1914 C. L. Stanton quotes Payne's concern "that if he struck a vessel with the torpedo staff projecting horizontally he feared the boat would enter the hole made by the explosion in the ship's side".  Payne's thought "that if the torpedo staff was lowered to an angle of forty-five degrees when the ship was struck the torpedo would explode near the keel, and the Fishboat's bow, striking the solid planking of the ship, would recoil sufficiently to make the machinery effective in backing out of danger".  Stanton continues "I have always felt very certain that the torpedo staff was in this position when the Housatonic was struck....  Besides, when the boat lay alongside the Chicora on the night of 14th of February, I examined it closely".  Housatonic crew accounts of the sinking indicate the Hunley was backing off at the time of the explosion, consistent with a lanyard activated trigger like Gift's. Chapman's painting shows what appears to be a spool for such a lanyard.
     The above accounts indicate the angle of the torpedo spar was adjustable.  Alexander gives this account a typical Hunley sortie: "Dixon and myself would ... get the bearings of the nearest vessel as she took her position for the night; ship up the torpedo on the boom, and, when dark, go out, steering for the vessel, proceed until the condition of the men, sea, tide, wind, moon, and daylight compelled our return to the dock; unship the torpedo, put it under guard at Battery Marshall..." (from Virgil Robinson's Southern Historical Papers page).  McClaurin reports that on Feb. 17 "Lieutenant Dixon landed and requested that two of my regiment ... go aboard and help them to adjust the machinery, as it was not working satisfactorily. Another man and I went aboard and helped propel the boat for some time while the Lieutenant and others adjusted the machinery and the rods that held the torpedo and got them to working satisfactorily".  Photographs of the semi-submersible David torpedo boats show a rig that permitted the torpedoes to be lowered below the surface near their targets.  Using such a rig would be difficult and dangerous from the very narrow hatches of the Hunley but McClaurin's machinery may have been intended for that purpose.  It is also possible that Dixon was making adjustments to keep the torpedo stable or at a particular depth.