Can we abandon conjecture and look at what singers and collectors have said about this song?

It has been collected three times from traditional singers, in 1952 for the BBC, from Jimmy McKee, Armagh (Folktrax); from Frank Mills, Tyrone by Robin Morton in about 1968 (published in Folk Songs Sung in Ulster); and in 1980 by Keith Summers & others from Patsy Flynn at Magheraveely, Fermanagh (as the Grangemore Hare)(issued in The Hardy Sons of Dan); in addition, the version recorded by The Irish Country Four may have some local authority, though the notes give no provenance.

I don't have access to the McKee version but Peter Kennedy and the BBC Catalogue have this to say:

From the Folktrax Catalogue
GRANEMORE HARE, THE - "One fine winter's morning our hunt they did blow" comp by Pat Toner of Tullyglish, Keady and Owen Mc Mahon of Tassagh, Co Armagh, fellow huntsmen in Keady & milltown Hunt Club - tune: "Villikens" - Hunt took place at Granemore Rock, Hayes Public House and featured a black hare - ROUD#2883 - MORTON FSSU 1970 #42 from Frank Mills - Cf ARTHUR BOND'S MARE - HARE OF KILGRAIN - OLD GREY MARE - WHITE HARE OF CREGGAN -- Jimmy McKEE rec by PK, Armagh 10/7/52: RPL 18409/ FTX-431 - Frank MILLS, Co Tyrone: MERCIER IRL-12 1970 - STEELEYE SPAN: B & C CREST-22 1970 & CS-12 1973 with Gerry Conway on drums, titled "Greenmore Hare" - IRISH COUNTRY FOUR: TOPIC 12-TS-209 1971 - KESTY: MODELLO 1979 / FTX-240 "Hills of Granemore"- LIVING TRADITION LTCD-001 1994 DERVISH (Group from Sligo) "The Hills of Greenmore" (from Whirling Discs WHRL-001 1993)

From the BBC Index:

GRANEMORE HARE, The

Singer:Jimmy McKee                                                                 4,10    18409
Armagh.   
10.7.52 (P.K - S.O'B.)

'One fine winter's morning, our horns they did blow …'

Song written by Owen McMahon of Tassagh to the well-known air 'Villikens'.   A hunting song typical of many such which are still composed in the district.   Some verses in it are sung by the hare!   Cf. 'The White Hare of Creggan' (18532) which uses the same air.   Cf. also Sam Henry Collection, No, 12: 'The Hare of Kilgain' (from Antrim).

Robin Morton, in Folk Songs Sung in Ulster gives no information about authorship but Coyle is mentioned twice:

Verse 7
It being so early, I stopped for a while
It was little I thought they were going to meet Coyle

Verse 9
I blame McMahon for bringing Coyle here
He's been at the same caper for many a long year
Every Saturday and Sunday he never gived o'er
With a pack of strange dogs round the Hills of Granemore.

My copy of Robin's book records some variations which I evidently heard in my misspent Belfast youth.

The version in The Hardy sons of Dan, has only a verse similar to the one immediately above; Coyle is mentioned only once.

In addition, Tom Moore and Allan Hampton give the song (mentioning Coyle twice) in their privately compiled book of hunting songs from county Down (Tom states that he heard it in the fities) and Seán McElgun gives a text, equally clearly from tradition but without any details in the second volume of his Songs of the Winding Erne (Hills of Graymore)- it names Coyle only once. Both give 'Coyle' with a capital letter, obviously, from the versions they heard, believing that it is a surname.

The surname Coyle is attested by Edward MacLysaght 'The Surnames of Ireland' and Robert Bell 'The book of Ulster Surnames' who associate it most frequently with Donegal, Derry, Tyrone and Cavan. It derives from an Irish sept name Mac Gilla Chomhghaill which was anglicised MacIlhoyle, MacCoyle and Coyle.

I think that conjecture and interpretation of songs is absolutely allowable but can see no justification in fact, context or syntax for the 'mortal coil' interpretation - but go ahead if you like, though for me the idea is no more than fantasy.

There is an old saying in Ulster which seems appropriate "Shall we let the hare sit?"