It depends on the area of Linguistics you are interested in - if it's an area which a large Psycholinguistics component, your current background may be enough. Tbh, Humanities PhDs are competitive funding-wise, so a lot of research councils ask for specific previous degrees - you might want to research potential supervisors and funding sources to see whether the route is viable for you.
There is also the MA+PhD route (1+3), which would fund a relevant masters degree plus the PhD, which might be suitable for you if you have a less than ideal academic background for Linguistics.