Olive oil--any food oil--should be fine. Bacon grease or Crisco even. Seasoning will cause a lot of smoke no matter what you use. Is this carbon steel? I only have experience with cast iron but I think they're pretty much the same for seasoning, so look into cast iron methods too. Some oils work better than others, or at least at different temperatures, but it sounds like you got the bottom seasoned nicely. If you have a grill outside and weather isn't awful, you might want to do the rest of it out there! The oven is good too, except for the smoke. Stovetop if you have wood handles though!

Once you burn in all the oil and get it seasoned, you won't have the smoke issues. If there is some gumminess to it, you might want to scrub that off before more seasoning, though.