It's fair to say that they could, if they want, head into April with the current crop, but now that the premium purchase aisle is cleaned out (apart from Bellinger) but there are a couple of aspects that would seemingly benefit from further tweaks. Let me take the small one first.
They need to decide whether Barger is an everyday guy who needs to not be sheltered from lefties so he can refine his production against them. He had years in the minors where his splits were fairly balanced so that would be reasonable and, frankly, what I want to see because I'm a committed believer in Addison Barger. BUT a case can be made to add some layer of insurance by seeking out someone who cruses lefties to get some reps in those situations. There's not a lot to choose from available across the league, but there are two (at least) remaining free agents who have been very productive vs lefties the last few years: Autin Hayes, and Miguel Andujar. Neither profile as a RF but you can juggle around to make it work. However, if you add a guy like that then Nathan Lukes (most likely) is squeezed off of the roster. It's difficult to discern how the team feels about this one.
The other is the question of second base. In my view, Ernie Clement is at or very near his ceiling and that ceiling is at best league average hitting, mostly against LHP with plus defense. If he's your everyday 2B then you're content to carry two hitters every day for their defense. There's no one better than him left in free agency so to upgrade here is to make a trade and, to be clear, any trade that's a notable upgrade will hurt you in the prospect costs.
Given that Mike Hazen in Arizona has declared the talks for Ketel Marte are now in the past, I submit that there are four possibilities.
I'll mention Nico Horner first as it seems unlikely he'd be flipped because of Bregman's arrival. Technically, you could try to change their mind if they think they'd be fine moving Shaw over, but For completeness, he's a marginally above-average hitter who, so the metrics say, is an excellent defender. That has added up to an impressive 17.5 fWAR over the last four seasons. Also, he's a free agent next winter in a really thin market so the heavy price would get you only one year, even if.
Another guy who's been displace by a free agent signing is Mets infielder Brett Beaty. A 26-year-old LH hitter who can play second and third, has 4 years of control, who put up a 2.3 fWAR last year. He split a bit better vs RHP last year (comparable to Barger in that regard) so you could work in Erine vs LH at both 2B and SS. His major league track record id shorter but he's a former first rounder who smoked the ball in the minors. He'd probably cost marginally less than the next two in terms of prospects but one would think that the Mets are not interested in just dumping him either. You have the benefit here of them not trying to ask for Barger in return since they wouldn't have an opening at either 3B or RF. Worth exploring.
The guy who's been talked about the whole off-season is Brendan Donovan. He's also a LH hitter, with two years of control remaining and who has a track record of 2.5-3 WAR on average each year. His splits lean heavily towards hitting RH setting up a theoretical platoon with Clement. But the reporting all winter has been the asking price is high and multiple teams are interested so you'd do well just to get him without trading someone (other than maybe Lukes or Schneider) off of the major league roster. Even then, can the Jays outbid the competition at a price they are willing to pay for just two seasons?
Which brings me to my favorite target: C.J. Abrams. He's the Nationals' full time SS although the defensive metrics are not a fan. If you land him then he's going to 2B. He's 25, has three years of control, a LH hitter who's career splits lean towards the direction you'd assume. By bWAR he's been 3.4 or better the last three seasons who can hit 20 homers and steal 30+ bases. What would he cost? Presumably A LOT. Some would ask "Why would they consider it?" to which I reply - if they'll listen on Mckenzie Gore they'll listen on Abrams
As with Donovan, if someone like Seatle found out he could be had they would rush to join the bidding so it's not just a matter of paying what he's worth but making a better offer than any other team has. I have no real idea if the Jays can do that without a very harsh price, but for the purpose of this speculation I'll rule out dealing anyone on the major league roster other than Nathan Lukes or Davis Schneider. (I can imagine a scenario in which they sign Cody Bellinger with a view towards including Barger in a trade like this but I'm not going to get that complex.
As an example of what the price would look like, assume a rubric like this: We have a basket for guys who are still rookie eligible that you can't have. To start Trey Yesavage is already in it. Next, between Arjun Nimmala, JoJo Parker and Juan Sanchez - you are free to pick one but the other two go in the basket. Next, you may choose between Ricky Tiedemann, Gage Stanifer, and Johnny King - the other two go in the basket. Finally, we'll add pretty much any two other guys not already in the basket. Imagine something like Fernando Perez and Josh Kasavich but you can't entirely know who they might find appealing.
Would I trade Nimmala and Stanifer and Perez and Kasavich for three years of Abrams? I think so. Would the Jays? Quite possibly. Would it be enough? I have no idea. But I'd spend the next month or more pushing them.
You could slide him (or Donovan) right into the #2 slot in the order and have an awfully well-rounded lineup. I don't have a clue whether anything like this would or could happen, but I can't believe that Ross Atkins feels fully content to play Clement at 2b virtually every day next year and not see it as a position that needs an upgrade.