The phrase "almost persistent" is correct and usable in written English.
It can be used to describe something that is nearly or close to being persistent, but not quite fully so.
Example: "His almost persistent attempts to reach her were both admirable and frustrating."
Alternatives: "nearly constant" or "borderline persistent".
The ailing infants, some near death, began arriving daily at Israeli hospitals in early November with almost identical symptoms: persistent vomiting, spasms, eyes that could not focus, listlessness.
But I am going to add to the word-count, because I think there is a persistent, almost willful, misunderstanding going on regarding the main points of the editorial.
Pay special attention to the sub-personalities that are persistent, almost desperate.
Those who start out poor not only continue to hit "normal" intractable barriers to greater economic opportunities, they also have to deal with persistent, almost undetectable discrimination - at least undetectable to the people doing the discriminating..
The Treasury, which is hoping a full-blown recovery is under way after almost two years of persistent economic weakness, seized on the widespread nature of the expansion signalled in the figures.
BALTIMORE — A disappointed Mark Teixeira said Monday that his persistent and almost debilitating cough had not improved the way he had hoped, and that he was at a loss trying to figure out how to get rid of it.
He had a "persistent and almost predatory interest" in girls, the court heard.
Inflammatory cytokines play a key role in fibrosis, given that persistent inflammation almost always precedes fibrosis [42].
Mohammed Khubaib, 43, had a "persistent and almost predatory interest" in teenage girls, the Old Bailey was told.
A married restaurant owner with a "persistent and almost predatory interest" in girls has been found guilty of rape and grooming offences.
