Accessing Private Property to track or retrieve game

Penal Law Section 140.05​

Trespass​

A person is guilty of trespass when he knowingly enters or remains unlawfully in or upon premises. Trespass is a violation.

S 140.10 Criminal trespass in the third degree.
A person is guilty of criminal trespass in the third degree when he
knowingly enters or remains unlawfully in a building or upon real
property
(a) which is fenced or otherwise enclosed in a manner designed to
exclude intruders; or
(b) where the building is utilized as an elementary or secondary
school or a children`s overnight camp as defined in section one thousand
three hundred ninety-two of the public health law or a summer day camp
as defined in section one thousand three hundred ninety-two of the
public health law in violation of conspicuously posted rules or
regulations governing entry and use thereof; or
(c) located within a city with a population in excess of one million
and where the building or real property is utilized as an elementary or
secondary school in violation of a personally communicated request to
leave the premises from a principal, custodian or other person in charge
thereof; or
(d) located outside of a city with a population in excess of one
million and where the building or real property is utilized as an
elementary or secondary school in violation of a personally communicated
request to leave the premises from a principal, custodian, school board
member or trustee, or other person in charge thereof; or
(e) where the building is used as a public housing project in
violation of conspicuously posted rules or regulations governing entry
and use thereof; or
(f) where a building is used as a public housing project in violation
of a personally communicated request to leave the premises from a
housing police officer or other person in charge thereof; or
(g) where the property consists of a right-of-way or yard of a
railroad or rapid transit railroad which has been designated and
conspicuously posted as a no-trespass railroad zone.
Criminal trespass in the third degree is a class B misdemeanor.

You are correct.
I had a pretty bad situation 3 years ago.
Like an idiot, I sat way to close to my neighbors property. I have a LOT of land, but I chose to sit where I did.
First light of opening day I shot a huge 10 point buck. The shot hit a little high, and the deer ran off down a hill, onto the neighbors property. I followed the deer about 50 yards onto the neighbors HEAVILY posted property.
There was a 6 foot wide spray of pink foam, but no deer.
The 80 something year old neighbor leases his property to a hunting club from Pennsylvania. I went down to where the hunting club guys have their camping trailers but there was nobody there.
I went to the land owner, but he said his hands were tied, and he could not give permission. I went back to the trailers and waited for someone to show up.
About 2 hours later a guy showed up. He said absolutely "NO, we pay a lot, you cannot get your deer!"
I was not sure of the rules, so I called the DEC. They sent an officer. He very politely asked if HE could go retrieve the deer. The lease holder said NO.
The officer made a phone call. He came back and said that he believed there was an animal in peril, and may be suffering.
Another DEC officer showed up, and they went and retrieved the deer for me.
Maybe kind of a dick move, but I got my deer.
The old couple sold their property (360 acres) this last Spring. The new owner did NOT renew the hunting clubs lease. He told me to absolutely get an injured deer if it goes on his property.

Yes you are correct and you need to get permission.