Whoa! Define metabolites/constituents. What do you expect to find? What are the expected concentrations?
Just because you have a 'super duper ultra' analytical instrument does not mean you don't have to do your research reviewing what other people have found!
Don't be surprised that you will see 100's of peaks when you make an injection. In addition, if you can smell an essence at room temperature means it is volatile and its concentration decreases with time. Thus, you will have to use a cyrogenic cooler attached to the GC in order to achieve stable results. You cannot escape a molecule's chemistry with GC/MS/MS.
Also use a 'headspace analyzer (HS)' since this will inject only the volatile compounds. Fatty acids and phospholipids will foul the GC column and kill it.
Most of the essential oil components are volatile in nature and can be analyzed by GCMS. You can do a direct liquid injection using an auto sampler OR as suggested by Bruce, use a Head Space sampler along with the GCMS which will ensure that only volatile components are going to the column. In GCMS you have the freedom to do library search.
If you decide to use GC-MS, as it was correctly suggested above, try splitless/on-column injection for less volatile compounds of your sample. If you will try to use headspace, if possible combine it with SPME, this will allow to enrich your sample. Alternatively try to use a dynamic hedspace/purge&trap system.
If your chromatogram will be to "rich", you can separete your sample and analyse fractions of different polarity. For first test you can try to use SPE, for example C18 eluted with a sequence of eluents in order of decreased polarity.
The analysis of metabolites will be more chellenging. Some of them will not be so volatile, thus you can think about derivatisation and GC-MS, or make some screening tests with HPLC. UV-DAD detector can be useful to find oxidized metabolites. At the end you can collect a fraction of interest from HPLC and further analyse it by GC-MS (same as for SPE).
The GC/MS databases from NIST and Wiley for electron impact spectra are said to be helpful, with a couple hundred thousand compounds.
Sounds like some good answers from other people, above. Simulated distillation (thin film DB-1) columns from Restek can go to very high temperatures, if you are interested in the heavier constituents. I used to favor the simulated distillation columns from Restek, silicosteel, for analysis of heavy hydrocarbons and vegetable oils.
For volatiles, you might consider a purge and trap apparatus. This could bump your sensitivity up hugely, for metabolites. Essentially all of the volatiles in a 5 gram or so sample end up on your column, leading to extreme sensitivity, with a purge and trap.
Because essential oils are volatile and thermally-unstable, their chemical composition is investigated by GC-MS. LC-MS is used to analyze even least volatile or thermally-unstable compounds that are difficult to analyze using GC-MS. LC-MS is highly sensitive, accurate, and specific compared to HPLC.