There are several Canon races that use different Psionic techniques than the Limited ones provided for the Betazoids and Vulcans. Putting aside those that would seem most like those – the Remans abilities for example, there are others that are pretty interesting. I’ll give you some samples and links in Memory Alpha so you can check them out for yourself.
The Ullians: Divergent from basic Telepathy, they people deep dive into a subject’s memories. TNG era. https://memory-alpha.fandom.com/wiki/Ullian
The Napeans: Relatively recent in their history, a bioterrorism act rendered them Empathic Projectors… they have trouble not involuntarily projecting their emotions, particularly the negative ones. The intent was to make them stop warring with each other as the victim’s pain was projected into the attacker. TNG era. https://memory-alpha.fandom.com/wiki/Napean
The Cairn: A strong Telepathic species but unlike the Betazoids and Vulcans, one that didn’t use verbal communication at all and instead had a massive library of Pictographs/Memes they sent to each other as language. The Betazoids were helping them learn to vocalize internally and externally after they made contact with the Federation. TNG era: https://memory-alpha.fandom.com/wiki/Cairn
Gary Mitchell and Elizabeth Dehner: Two Humans who had high ESPER ratings, the actual canon term used. Mostly this reflected a mild precognative and/or clairvoyant ability, but after they were taken through the Galactic Barrier they powered up into beings with so many powers and at such a level that they could literally terraform the planet they were on with their minds. TOS era: https://memory-alpha.fandom.com/wiki/Gary_Mitchell and https://memory-alpha.fandom.com/wiki/Elizabeth_Dehner respectively.
In the Last Unicorn Games and Decipher takes on Star Trek, there was an additional Psionic attribute that governed such things. In a home grown game you could easily add this… give a base stat level to species that come by their Psionics honestly as a race, allow those from races that have potential but aren’t fully that way to spend a talent on it. You could then add a short list of Psionic Disciplines: Telepathy, ESP, Psychokinesis, etc and then add Focus for these Telepathy, Psychometry, Telekinesis etc – all with GM permission limitations, and treat them like the other mundane mechanics.
Picking up a book like: https://www.amazon.com/Way-Kolinahr-Vulcans-Generation-Unnumbered/dp/0671040081 Will provide you with great flavor to such a thing, this one covers the various ESPer arts of the Vulcans in nice thematic appropriate detail.
Or you can just tell your player that its not something you want to deal with and focus on either things or let them spend a Talent on something called Psychometry and have them make an Insight+Command test to use it and drive on. The important thing is, such an ability is thematically appropriate so you don’t have to worry about including it in your game if you want to. Just don’t let the character hijack all your mystery plots by touching an item and saying "I know who the killer is: – Psychometry should be showing what an emotional residue would reveal not a security feed. YMMV, Peace.