What weight oil are you running

I just had my 340 rebuilt. Was wondering is 10w-40 still ok? Im gonna put zink additive in as well. This is all after the break in. I have break in oil for that.
do you have the bearing clearance specs?
This ^^^^^^^^

Here's what was posted in:
Oil Viscosity vs Bearing Clearance
and discussed in posts here
Valvoline VR-1 vs. Lucas Hot Rod oil
Your engine builder may have some thoughts on this too.
For street & strip use the following SAE grades:
Rod bearings with
.0015 to .0018 use 20
.002 to .0024 use 30
.0025 to .0029 use 40
.0030 to .004 use 50

For main bearings:
.0015 to .002 use 20
.002 to .0025 use 30
.0026 to .003 use 40
.0031 to .0041 use 50

For general use, its usually better to use a multi-grade.
whateverW Same will provide better lubrication when the oil is not fully warmed up than a straight grade.

Some recommended rod bearing oil clearances for various oil viscosities:
.0015 to .0018 for 0W20 or 5W-20
.0020 to .0024 for 5W-30 or 10W30
.0025 to .0029 for 10W-40 or 15W40
.0030 to .0040 for 15W-50 or 20W-50

For main bearings:
.0015 to .0020 for 0W20 or 5W20
.0020 to .0025 for 5W-30 or 10W30
.0026 to .0030 for 10W-40 or 15W40
.0031 to .0041 for 15W-50 or 20W-50

After break-in, depending on how it was machined, probably best to use a conventional base stock for a few thousand miles.
One current option for a conventional with decent Zddp levels is "Ford Super Heavyduty" meets FORD WSS-M2C171-F1
PETROLEUM QUALITY INSTITUTE OF AMERICA TEST PROGRAM