How did Bligh navigate during his open boat voyage after losing The Bounty?

The book The Eventful History of the Mutiny and Piratical Seizure of H.M.S Bounty by Sir John Barrow includes a chapter on the remarkable voyage of Bligh and his 18 companions in their 23-foot boat, drawn from Bligh's description of the voyage. Unfortunately, Barrow concentrates on the hardships of the men and includes few details on how navigation was accomplished.

Bligh's own, more lengthy account, was given in A Narrative of the Mutiny on Board His Majesty's Ship Bounty which was published in 1790. This includes details of the mutiny itself and the subsequent voyage of Bligh and his companions.

As far as the navigation tools available, Bligh states that as he and his companions were placed in the boat, the boatswain and seamen were allowed to collect items to take on the voyage.

Mr.Samuel [ship's clerk] got 150lbs of bread, with a small quantity of rum and wine. He also got a quadrant and compass into the boat; but was forbidden, on pain of death, to touch either map, ephemeris, book of astronomical observations, sextant, time-keeper, or any of my surveys or drawings.

^{A Narrative of the Mutiny on Board His Majesty's Ship Bounty, pg5}

It's not entirely clear whether the "compass" referred to in the passage, was a magnetic compass or a divider caliper. However, given that they had no charts, the latter would seem to be of little use. Also Bligh's log has regular entries on wind direction and direction of travel so the presence of a magnetic compass seems more likely. The quadrant was used to observe latitude and while longitude was taken 'by account' (i.e. by dead reckoning).

[May 9th] At noon, I observed the latitude to be 15°47'S; course since yesterday N 75°W; distant 64 miles; longitude made, by account, 8°45'W.

^{A Narrative of the Mutiny on Board His Majesty's Ship Bounty, pg49}

When the weather prevented an observation, then latitude was also taken by account.

[May 12th] At noon it was almost calm, no sun to be seen, and some of us shivering with cold. course since yesterday W by N; distance 89 miles; latitude, by account, 14°33'S; longitude made 13°9'W. The direction of my course is to pass to the northward of the New Hebrides.

^{A Narrative of the Mutiny on Board His Majesty's Ship Bounty, pg53}

In addition to the navigation tools already mentioned, it's implied that Bligh used the stars to navigate since the account mentions a couple of occasions when the poor weather made it impossible.

The whole day and night of the 15th were still rainy; the latter was dark, not a star to be seen by which the steerage could be directed, and the sea was continually breaking over the boat. On the next day, the 16th, was issued for dinner an ounce of salt pork, in addition to their miserable allowance of one twenty-fifth part of a pound of bread. The night was again truly horrible, with storms of thunder, lightning, and rain; not a star visible, so that the steerage was quite uncertain.

^{The Eventful History of the Mutiny and Piratical Seizure of H. M. S. Bounty, pg104-105}

Often, because of the weather, they simply went where the sea carried them.

The night was a dark and dismal one, the sea constantly breaking over us, and nothing but the wind and waves to direct our steerage.

^{The Eventful History of the Mutiny and Piratical Seizure of H. M. S. Bounty, pg105}

As for navigating the barrier reef, it seems that they came upon this unexpectedly and handled the navigation of it by sight and sound.

Accordingly, at one in the morning of the 28th, the person at the helm heard the sound of breakers. It was the 'barrier reef' which runs along the eastern coast of New Holland, through which it now became the anxious object to discover a passage: Mr. Bligh says this was now become absolutely necessary, without a moment's loss of time. The idea of getting into smooth water and finding refreshments kept up the people's spirits. The sea broke furiously over the reef in every part; within, the water was so smooth and calm, that every man already anticipated the heartfelt satisfaction he was about to receive, as soon as he should have passed the barrier. At length a break in the reef was discovered, a quarter of a mile in width; and through this the boat rapidly passed with a strong stream running to the westward, and came immediately into smooth water, and all the past hardships seemed at once to be forgotten.

^{The Eventful History of the Mutiny and Piratical Seizure of H. M. S. Bounty, pg110-111}

Bligh knew from his earlier travels with Cook roughly where he was at that point and, therefore, knew the rough heading for Timor. He estimated correctly the boat's position 11 days later and was able to estimate the entire distance covered in the voyage -

On the 11th Lieutenant Bligh announced to his wretched companions, that he had no doubt they had now passed the meridian of the eastern part of Timor, a piece of intelligence that diffused universal joy and satisfaction. Accordingly, at three in the morning of the following day, Timor was discovered at the distance only of two leagues from the shore...It appeared scarcely credible to ourselves that, in an open boat, and so poorly provided, we should have ben able to reach the coast of Timor in forty-one days after leaving Tofoa, having in that time, by our log, a distance of three thousand six hundred and eighteen nautical miles; and that, not withstanding our extreme distress, no one should have perished in the voyage.

^{The Eventful History of the Mutiny and Piratical Seizure of H. M. S. Bounty, pg118}