We know Gandalf, Radagast, Saruman, Pallando and Alatar, who were all Istari, Maiar in physical form. The Elves can also do 'magic', as can Sauron, being a Maiar himself. Were there any other people with magical capabilities? The Nazgûl also seem to be able to do magic, the Witch-king made Frodo's sword break, for example.
Could wise men learn 'magic' by studying?
Depends what you mean by 'magic'.
The Numenoreans could certainly use "magic", as evidenced by their construction of Orthanc, creation of the magic blades Merry and Pippin wielded, curse of undeath placed on the dead men of Dunharrow etc.
The Pukel Men (Druedain) could create magic statues to attack their enemies.
The Ice Men of Forochel had prophetic powers (predicting the death of Arvedui)
So I would say yes, but the types of magic were different to "wizard" magic.
Well this quote from the chapter "The Black Gate Opens" in LOTR suggests that the Mouth of Sauron fits the criteria of your question (the emphasis is mine):
But it is told that he was a renegade , who came from the race of those that are named the Black Numenoreans; for they established their dwellings in Middle-earth during the years of Saurons domination, and they worshipped him, being enamoured of evil knowledge. And he entered the service of the Dark Tower when it first rose again, and because of his cunning he grew ever higher in the Lord's favour; and he learned great sorcery, and knew much of the mind of Sauron; and he was more cruel than any orc
No detail is given but it at least seems to me that he was mortal and learned sorcery under the guidance of Sauron.
Well Magic in middle Earth has 5 main categories, and different people access the different types of magic for different reasons.
Prophecy: the most common type, in the series it is used by Glorfindel, Boromir and Elrond
Shadow Realm: Creatures such as the ring wraiths can access this realm for power. Elves live in both normal and spirit realm at the same time. When wearing the One Ring, you can access this realm.
Magic items. Normally have vague stories about their making normally citing the Elves or a greater force.
Spells!! The most well known type of magic, can only be used by a Maiar/Istari.
Talking animals. Never really described.
"And you?" she said, turning to Sam. "For this is what you folk would call magic, I believe; though I do not understand clearly what they mean; and they seem to use the same word for the deceits of the Enemy. But this, if you will, is the magic of Galadriel. Did you not say that you wished to see Elf-magic?"
There are no recorded non-Istari wizards in Middle-Earth, but it really depends on what you mean by "wizard".
Before the occupant of Dol Guldor was identified as Sauron, he was called "The Necromancer", which could be described as a wizard that practices death magic, and communing with the dead.
The Mouth of Sauron was described as a Sorcerer.
Boromir notably described Galadriel as "A great sorceress", prior to their meeting.
However, it should be noted that none of these individuals actually performed any overt magic. Most magic in Tolken's world was far more subtle, and described more as a natural art than conjuring. If anything, magic in Middle Earth was more like enchantment, i.e. objects and people imbued with extra-natural properties, e.g. The rings, the silmarils, Galadriel's mirror etc.
However, I digress.
The fact remains that in the published canon, no there were no instances of wizards outside the five who came from the far west at the end of the second age.
The 9 Wizards of the Heren Istarion (“Order of Wizards”) 9 different colors
SARUMAN (Curumo) the White – sent by AULË
ELASTER (Otiluë) the Black – sent by NÁMO
GANDALF (Olórín) the Grey – sent by MANWË and VARDA
RADAGAST (Aiwendil) the Brown – sent by YAVANNA
BLADORTHIN (Pallando) the Blue – sent by OROMË and ULMO
GALDATHOR (Alatar) the Green – sent by OROMË
RENGOL (Aeral) the Red – sent by TULKAS
PERDUSTIN (Nólimo) the Purple – sent by IRMO
YANZEL (Morinehtar) the Yellow – sent by NIENNA
