The Mormon Immigration Index has the following:
HANCOCK, Henry <1814> Amazon 1863
Age: 49 Origin: England Occ: Shoemaker
Note: BMR, p.294
Ship: Amazon
Date of Departure: 4 Jun 1863 Port of Departure: London, England
LDS Immigrants: 895 Church Leader: William Bramall
Date of Arrival: 18 Jul 1863 Port of Arrival: New York, New York
Source(s): BMR, Book #1047, pp. 277-316 (FHL #025,691); Customs #747 (FHL #175,587)
Notes: "AMAZON. -- The splendid packet ship Amazon, Captain H. K. Hovey, also sailed from London on the 4th instant, with a company of 895 souls of the Saints on board under the presidency of Elder William Bramall; Elders Edward L. Sloan and Richard Palmer being associeated with him as his counselors. The company passed the Government Emigration Officers on the 3rd, who eulogized their order, harmony and general appearance, after which Presedent Cannon, who was accompanied by several elders from various parts of the mission, held a meeting organized the company and gave appropriate instructions. The interest manifested by strangers and the officials whose duty called them to be contiguous to the ship, evinced how much excitement the novelty of a ship-load of Saints, leaving London, produced. During the meeting which accompanied the organization, the officers of the ship, the cabin passengers and the visitors on board listened with marked attention; while the unanimity of feeling manifested by the Saints, and the deep interest with which they listened to the instructions given and took part in the proceedings on the occasion, evidently made a deep impression on them, displaying, as it did, a something so different from all their conceptions of us as a people. A brass band, from Soulth Wales, the performers being members of the Church on their way to Zion on the Amazon, discoursed sweet music on the poop-deck before and after the meeting, while the sun shone down upon the crowded deck as if the heavens and the earth were combining together to bestow their blessings upon the last company of the season. The presidency having been appointed and Elder William M'Lachlan nominated as clerk, Elder Kay closed the meeting with prayer, President Cannon having pronounced a blessing upon the ship, her officers and crew and the Saints on board. There was considerable excitement manifested by the people on shore as this vessel left the dock and moved down the river, the people on the wharves cheering, and, on the banks of the river and on the vessels anchored in the stream waving their handkerchiefs and hats and giving vent to other demonstrations in response to the singing of the people and the music of the band. It is worthy of note that the departure of the Amazon from London, laden with Saints, is another instance of the fulfillment of prophecy. Some years ago, while Elder Eli B. Kelsey was laboring in London, he predicted in a public meeting in that city that ships should yet leave that port filled with Saints emigrating to Zion. It was with no intention of bringing about the fulfillment of the at prophecy that we chartered the Amazon, for we were entirely ignorant of the utterance of such a prediction until we heard it stated in a meeting of the Saints held on Sunday, the 1st instant, three days before she sailed. Indeed, the chartering of this vessel was not a matter of choice with us but of necessity. We could not obtain a vessel in the port of Liverpool suitable to our pupose -- vessels of this description being almost unprecedentedly scarce this spring, and we were, therefore, compelled to go to London. Thus were circumstances overruled to bring to pass the fulfillment of the words of a servant of God! Of the elders who sailed on the Amazon, four were from the Valley -- Elders Bramall, Palmer, Edward T. Edwards, and A. W. Van der Woude. . . ."
<MS, 25:25 (June 20, 1863), p.395>
"June. Thurs. 4. [June 1863] -- The packet ship Amazon sailed from London, England, with 882 (or 895) Saints, under the direction of William Bramall. It arrived in New York harbor July 18th, and the immigrants reached Florence [Nebraska] a few days later."
<CC, p.69>
Also on the same ship were:
HANCOCK, Ann <1813>
Age: 50 Origin: England Occ: Wife
HANCOCK, Joseph <1813>
Age: 50 Origin: England Occ: Blacksmith
Note: BMR, p.301
HANCOCK, Henry <1814>
Age: 49 Origin: England Occ: Shoemaker
Note: BMR, p.294
HANCOCK, Sarah <1814>
Age: 49 Origin: England Occ: Wife
HANCOCK, Horatio <1843>
Age: 20 Origin: England Occ: Painter
Note: BMR, p.314
HANCOCK, Anne <1851>
Age: 12 Origin: England
HANCOCK, Eliza <1853>
Age: 10 Origin: England
HANCOCK, Ann <1854>
Age: 9 Origin: England
HANCOCK, William <1856>
Age: 7 Origin: England
Hope that helps
jds1949