Direct CO2 measurements of CO2 in Mauna Loa, HI (MLO) began in March, 1958. Regarding the 1958 CO2 level, the TL;DR summary is that MLO measurements and ice cores both showed the average 1958 CO2 level to have been about 315 ppmv†:
The 1959-2025 Mauna Loa (MLO) CO2 levels in my graphs are from NOAA:
https://gml.noaa.gov/webdata/ccgg/trends/co2/co2_annmean_mlo.txt
The 1850-1958 ice core CO2 levels and the 1958 MLO CO2 level are both from GISS:
http://data.giss.nasa.gov/modelforce/ghgases/Fig1A.ext.txt
GISS gives the 1958 CO2 level as 315.34 ppmv, and describes the 1958 & later levels as "SIO Mauna Loa & South
Pole Adjusted for Global Mean," which isn't strictly MLO. But the 1959 level which they report is nearly identical
to NOAA's 1959 MLO level (316.18 vs. 315.98) so I think that's a distinction without a real difference.
NOAA also has MLO measured monthly CO2 levels for March through December of 1958, here:
https://gml.noaa.gov/webdata/ccgg/trends/co2/co2_mm_mlo.txt
Averaging their deseasonalized values for March through October of 1958 yields 315.30 ppmv, which is a close approximation of the full year average level, and nearly identical to the 315.34 ppmv level reported by GISS.
The 1800-1849 ice core CO2 levels are the 75-year smoothed Law Dome ice core CO2 levels from NOAA, interpolated as necessary:
https://www.ncei.noaa.gov/pub/data/paleo/icecore/antarctica/law/law_co2.txt
They also report 20-year smoothed levels, but only back to 1832.
The NOAA and GISS figures sometimes differ by a few tenths of a ppmv.
There are links to those data sources, plus ice cores which go back much further than Law Dome, in the footnotes of the graph webpage:
https://sealevel.info/co2.html
You can also "view source" for that web page to see the data, which is embedded in the html file. For the CO2 graph,
the 1958 data is around line 500.
† “ppmv” means parts per million by volume, which is shorthand for µmol/mol of dry atmosphere.