I am looking at a Papuan language that uses a serialized verb to denote temporal proximity to the main verb's occurrence. I am translating it as "just" in English, as in "he just left", or "I am just going now". This verb works with both past and future tense, describing an action that is just about to happen, or something that has just happened. I am not sure how to describe this verb in linguistic jargon. It describes aspect, but what type of aspect? Hoping someone can help me with this.
I am not sure this answer responds to your question, but attempting an answer seemed better than prolonging a question-and-answer dialog through the comments.
The closest parallel I can think of to what you describe is the Mandarin word 来着/來著/láizhe. (By the way, some rules about word separation of Chinese words written in pinyin are finicky, so you might encounter this word written separately as "lái zhe.") If you Google "láizhe aspect," you will see a bunch of citations that might help you, including section 3.2 of this link.
If you are not familiar with Mandarin and do not want to set out on a wild goose chase, let me explain some of the usage of láizhe in case it is comparable at some level to your Papuan word. If it is comparable, you can research the Mandarin word further to see how it is described in terms of aspect. If it is not, you can ignore what I say.
Descriptions of Mandarin aspect are quite involved and differ in their treatment, so I am offering this fuller account in case it might give you guidance for your language. My abilities in Mandarin are moderate, so take what I say below with a grain of salt.
The definitions I can easily find for 来着/來著/láizhe seem to be completely inadequate and contradictory, but my sense is that the word is used to describe some action as having particularly recent application to the current state of affairs. Mandarin is usually described as tenseless, but with a heavy focus on verb aspect.
There is another word that is the normal equivalent of English "just," which is 刚才/剛才/gāngcái and which can co-occur with láizhe, but usually does not and is much more commonly encountered as the temporal equivalent of English "just."
I would not describe láizhe as being central to the TAM system of Mandarin, but it is a combination particle that does seem to express aspect, at least from a discourse perspective.
Here are some examples of the use of láizhe I found in one of my dictionaries that express nuances not covered in the Wiktionary link I provided above (I will use only simplified characters):
他去年冬天还回家来着。 Tā qùnián dōngtiān hái huí jiā lái zhe. He was home only last winter.
我进门时，他在这儿站着来着。 Wǒ jìnmén shí, tā zài zhèr zhànzhe láizhe. When I came in, he was standing here (just now).
你昨天让我买什么来着？ Nǐ zuótiān ràng wǒ mǎi shénme láizhe? What did you ask me to buy yesterday (again)?
At the surface level, láizhe functions as a combination particle, but it is fairly transparently made of two morphemes: lái, which means "to come," and zhe, which is grammaticalized morpheme usually used to make an expression stative. I don't think the aktionsart of lái would normally allow this combination on its own, but the flavor of the combination seems to be that the previous clause "comes into" or "leads into" the current discourse situation, which is otherwise undescribed.
Syntactically, láizhe seems to originate as serial clause (?) expressing the discourse focus, but not the substance of the expression.
The word order in Mandarin would not easily allow a sentence-final particle to refer to time prior to the verb in a previous clause and so cannot use a sentence-final particle to express "I am just going now, but I'm not sure that is relevant for your Papuan language. Mandarin also has a range of expressions to describe near future events, but I don't think láizhe is one of them, since this would also violate the word order rules and lead to a nonsensical meaning from a discourse perspective (e.g., "You are recently going to do something").
Although the Google citations I referred to above talk about láizhe as an aspect particle, treatments of aspect in Mandarin vary widely in terms of what are treated as fundamental aspects and this particle is definitely somewhat peripheral to the basic TAM system. My feeling is that it is more of a temporal adverb or discourse particle than an aspect particle. In Mandarin, there is consideral overlap between TAM particles and discourse particles.
Using Mandarin as my perhaps inadequate guide, I would think how you describe the Papuan particle would depend on your treatment of the rest of the language. If you treat it as an aspect particle, then I would expect it to be one of a dozen or more for such a specific meaning. If you treat it as as a discourse particle, then you need a discourse theory for the language. If it is merely a particle helping narrow down the TAM reference, then it is an issue of defining its lexis.