Heads for mild 360 build

I’m looking to build a mild to moderate 360 build soon. Has an aftermarket cam(unsure what it is because previous owner installed it) new airgap intake, stock (J) heads to that 360, headers, 650 cfm carb. Couple other things. Looking for around 350+ when it’s put together. I want it to be a nice daily street car that can also whoop most cars on the street

My question is, what heads would you recommend for my situation. And even new camshaft and valvetrain possibly.
I usually wait until page 2 rolls around before I post, so that the infighting can wrap up.
What everybody said is probably right.
However, if you are at altitude, say over 4000ft and want an honest 350hp, IMO everybody is wrong. And the reason is that unless you get the Scr up, she will be a lazy stinking dog off the line and all the way to about 3000rpm., and then be a lil less lazy for the rest if the way. EDIT; Ok let me rephrase that for Rusty; "she will be a lazy stinking dog off the line" should read; she will PROBABLY be a lazy stinking dog off the line
I strongly urge you to find out more about that cam. The most important spec to try and find, is the Intake Closing Angle. And, you gotta ascertain the local elevation at which the engine will be operating. Without these two data points, I fear that you will be in for a big disappointment.
As for the 71 block going to 4.04, mine did, and by it's trapspeed of 93 in the Eighth, is making about 430 hp.......... at ~900ft elevation. This build was pretty "basic".

If there is no way of finding the cam-spec, then you will have to measure it, which really is not that big a deal.