Not pursue. One strike. Even if they only figure out where the fleet is based on reports from the fighters over Pearl radioing in what direction the attack came from, it would set the KB up for one beautiful sucker punch. The minute Halsey gets the rough bearing, he launches. Then, one of the land based fighters trails the Japanese back to their fleet giving a much better location for Halsey. The USN hits them hard, concentrating on only the 2 or 3 largest carriers, then gets the hell out of there.
Unless you (somehow) manage to put Halsey somewhere north or slightly northeast of Oahu & less than 175nm from Nagmo's launch point, it's going to be a pursuit. And he's very unlikely to get that "sucker punch" when Nagumo is expecting a counterattack (which he is). You might (just) make it work by using AAF a/c, with more range, but they'd have to
find Nagumo's TF, which IMO is improbable, since they're not trained for that...& any "trailer" is likely to be shot down by IJN escort fighters.
If (& it's a pretty big if) you can have PBYs detect & accurately locate Nagumo (& how you do that when Kimmel's only got a comparative handful of them & is expecting attack from a different direction, I have no clue), you just might get away with AAF counterattck--but now, IMO, you're pushing ASB.
And here's the bigger issue: Nagumo is
ready to attack CVs (unlike at Midway, where he got caught switching weapons; that
will not happen here). If he detects Halsey at sea,
anywhere, the chances are excellent Halsey has his ass handed to him, because Nagumo's fliers are all combat experienced, & none of Halsey's are. Nor are they really ready for how good the A6M is. It's likely, IMO, the U.S. loses
Enterprise &
Lex (maybe not, if there's reinforcement by
Hornet &/or
Wasp), tho the aircrew are likely to be able to recover to Oahu, unless contact is made further out. Losses to technicians & supporting personnel aboard the CVs will likely be heavy, too.
I hate to harp on it, but the best outcome is just about the OTL one. It may not look like it, at a glance, but it really was pretty good for the U.S., in all.
Imperial Japan had at least one spy at Pearl Harbour - Takeo Yoshikawa, according to wikipedia.
Does your point of departure assume that he is picked up and/or turned?
If not, what obvious signs of preparations to receive an attack does he (or other Japanese sympathetic to Imperial Japan in Hawai'i) see and report, and what do the Imperial Japanese Naval headquarters (in the time available) decide and do when they receive said reports?
Japan really only had him alone; there were
Nisei &
Issei, but they were ignorant of what he was doing, not complicit. Truth is, you don't need to pick him up or turn him to uncover the notion Oahu is a target; the amount of message traffic out of the Hawai'i station (consul?) was way out of proportion to what it should've been, & J-19 held clues--between the two, it wouldn't have taken
Kreskin to predict an attack.
Would he see more warships in harbor? Yes. More AA guns deployed? Yes. More troops in the streets more of the time? Yes. More fighter patrols? Probably.
Would IJNHQ change their plans to attack? I really doubt it. They expected to achieve surprise, expected things to go perfectly, to the point there was actually poor co-ordination between TBs, level bombers, & DBs, & poor target allocation. (I came across a PDF online criticizing the claim the plan's "brilliance"; good luck finding it, since I can't recall the web address or author...)
One thing about Yoshikawa, tho: if he sees increased preparations by the U.S., does he wonder
why? More important, does
IJNHQ? Does it lead them to suspect JN-25 is compromised? (Are we presuming it is?) If so, do they
change it? Also, if the U.S.
has broken JN-25 (not true OTL), does that lead to a broader warning to USG officials not to screw around with Japanese cyphers of any kind, for fear of blowing the gaff? If
that happens, there's a reasonable chance the
maru code is never copied by an idiotic Customs official

& so is
not changed.

(The benefit of this to the Sub Force once the war starts outweighs the harm done by the Mark 6 & Mark 14.)