LMSC A lot depends on what you want to achieve. If you just want to protect, good quality linseed or tung oil will do the job.
If the wood has gone grey and you want to return colour, then you may need to do other things, including sanding and staining; many stains will not work on top of cured poly, so beware the order in which you put coats and/or which products you use; the effect of a gel stain (that will work on top of varnishes) is very different - to my eyes, at least - than a typical penetrating stain, and it may not replicate what you originally had.
Also, varnish-based finishes like poly are great… while they last; then the cursing begins as they will eventually flake and removing a failed finish can be a pain in the @rse. Oil-based stuff will protect less, but is far easier to renew/maintain.
Lastly, a note on “oils” and ‘teak oil’ in particular: the name ‘Teak oil’ covers a multitude of sins; some ‘teak oils’ are very nice mixes of natural oils (including linseed and tung, but never oil from actual Teak!), others contain significant amounts of varnishes which may not actually be a good idea for your application… (or it could, depending). For that matter, the same applies to ‘tung oil’ - a lot of stuff sold as such contains very little tung oil. This article on Popular Woodworking goes into the ‘teak oil’ myth in a fair amount of depth https://www.popularwoodworking.com/finishing/teak-oil-what-is-it/
No matter what you end up deciding, I would make sure that the materials you use are of excellent quality; a lot better to have 2-3 coats of high quality finish, than slapping on 10 coats of stuff that is going to peel off after (or during…) the first season.