Just for a change, I would like to propose something that might easily have happened. All it would have taken is slightly more urgency in getting Kaga home to be repaired and Yamamoto trying to confuse his USN opponents.
OTL, Yamamoto had offered Kaga as support for Operation MO, which was aimed at the occupation of the Solomons and Port Moresby, but seems not to have made too many efforts to have her ready until an order was sent on 8th April http://www.combinedfleet.com/kaga.htm
. If we assume that Yamamoto had been more enthusiastic about MO, we could imagine that he ordered Kaga home on 6th March and told Sasebo to prepare to dock her as soon as possible. Ordering her home early gains a week and another five days can be gained if Kaga can enter the drydock immediately on reaching Sasebo. With rather more urgency, it seems likely that Kaga can complete her repairs by about 20th April, which is just too late to chase Enterprise and Hornet after their raid.
Yamamoto's plan for the attack on Midway has occasionally been subject to mild criticism for unduly dispersing his forces
but it does have a certain crazy logic. The logic, totally false, was that the USN would not dare to fight a battle unless they believed that only a fraction of the IJN was close to Midway. Thus he probably calculated that if the Americans knew that attacks were being made simultaneously on Midway and the Aleutians and if his battleships were hidden far enough behind the attacking carriers that they could not detected, the USN was more likely to come out and fight.
So what if Yamamoto had seen the operations against the Solomons and Port Moresby, Operation MO, as a further opportunity to convince the Americans that his forces were dispersed all over the Pacific. He would have known that there were a fair number of long ranged aircraft based in Australia which would report any IJN forces sailing in the Coral Sea. In OTL Inoue had demanded better support than Kaga and Yamamoto had agreed to detach the 5th Carrier Division as a stronger replacement. Thus after returning from Operation C in the Indian Ocean, Shokaku and Zuikaku had refueled at Mako and departed for Truk on 19th April http://www.combinedfleet.com/shokaku.htm
. They arrived at Truk on the 25th April and only departed to support MO on 1st May. In this universe, all the OTL forces taking part in MO initially act more or less as OTL but another force leaves Japan around 22nd April consisting of Kaga, Nagato and Mutsu with a destroyer screen and probably a light cruiser. This force will also need at least one and perhaps two oilers, which is probably the most disruptive feature of the suggested divergence for the Midway operation.
Possibilities for the light cruiser seem to include Jintsu or Naka and the destroyers to sail with either ship should logically be those already working with that cruiser. However, those two cruisers carry rear admirals, Tanaka Raizo on Jintsu and Nishimura Shoji on Naka. Mutsu's Captain Kogure was promoted to rear admiral on 1st May OTL but he will be junior to either of the destroyer leaders.
None of those three rear admirals will disrupt the lines of command of MO and any of them could be reasonably imagined as commanding the force. However, there are significant implications in Yamamoto's choice. Kogure is the blackest of black shoe admirals having spent most his career on battleships with some time on heavy cruisers (he commanded Chikuma before Mutsu). Hopefully, he will defer to Captain Okada Jisaku of Kaga over flying operations but he also knows nothing about destroyer tactics. Tanaka is present in Japan but is presumably preparing for his role in the Midway Operation. Nishimura is somewhere near Java and thus is hard to instruct unless he is flown to Japan and back again to his command. 21St Century readers will have very different expectations of the three admirals competence but we do not know how Yamamoto regarded Nishimura's performance off Balikpapan. If forced to guess, I would bet that Yamamoto would chose Tanaka on the assumption that he could quickly read the orders for MI when he returned to Truk, which would fit the way MI was planned. If the IJN was being sensible, they would try to keep Nishimura and Shima apart but I doubt if that would be considered.
Thus Tanaka's force will arrive at Truk after Takagi's but will surely be ready to leave with Takagi on 1st May. Takagi has to travel towards Rabaul to fly off aircraft, so Tanaka's force could leave slightly later, sailing towards a meeting with Takagi's force somewhere north of Tulagi before entering the Coral Sea. Of course, they will actually arrive ahead of Takagi because Takagi and Hara will run into bad weather which will delay and then prevent their sending the aircraft to reinforce Rabaul. Tanaka's force will probably refuel their destroyers while waiting for Takagi but it is hard to know where they will be when news arrives of an attack by carrier aircraft the ships at Tulagi.
I am assuming that the Allied response to MO exactly follows OTL. That is reasonable because up to five IJN carriers were expected from the code breaking OTL. In particular, Kaga was expected to take part along with Carrier Division 5 due to the early messages allocating Kaga to Inoue's 4th Fleet. Nimitz's acceptance of those odds does show up the error in Yamamoto's calculation. Thus we can expect Yorktown to launch its attack against Shima's ships on 4th May. OTL, Yorktown launched three strikes, the first was launched early in the morning and the last was recovered in the evening.
What happens next depends on where Tanaka's force is waiting on 4th May. It seems likely that they will be south and east of Takagi, who was roughly 400 miles north of Guadalcanal when informed of Yorktown's first attack. Will they be near enough to find Yorktown by evening? Will their returning aircraft be observed by Yorktown's radar and give Fletcher a clue to the position of the enemy carriers? Will Kaga's aircraft search the area east of Guadalcanal where Hara's aircraft searched OTL and naturally find nothing as Fletcher was south of Guadalcanal?
We could continue assuming that Takagi follows his OTL schedule with Tanaka a little ahead but perhaps there will be changes. Takagi was limited OTL to below 20 knots by his oilers https://en.wikipedia.org/wiki/Kawasaki-type_oiler
. However, with Tanaka ahead, he might have left his oilers behind and planned to steal Tanaka's fuel with the advantage of seniority. He might even be instructed to do that by Inoue, who could organise a rendezvous without either breaking radio silence. Even if the three large IJN carriers end up close to where Shokaku and Zuikaku were OTL on 6th May and not further south, they will be fairly close to the American carriers at dusk. With more aircraft searching, it seems possible that they will make contact. If there is contact near dusk, there is the possibility of a surface action. OTL there was 70 nmi between the carriers at 2000.
If no contact occurs on the 6th, we can imagine the blind man's bluff of the 7th and 8th going as OTL except that Takagi has more aircraft available to search or attack his opponents. The USN does have an extra target but there will be more Zeros. If there are at least two decks available when the main IJN strike of 8th May returns, fewer repairable aircraft will be tipped off Zuikaku's deck and it seems likely that a second strike will be launched during the afternoon. However, it is more likely that events will diverge before then.