Lee,

I'd start by gathering two pieces of info:

1. every census that your grandfather will appear in should have a column asking either "How long has this person been in the US" or "What year did they arrive in the US".  Many states (especially NY) did their own censuses in between federal census years and also asked this question.  From the answers, you should be able to get a good approximation of the year that your grandfather arrived.

2. if any of your foreign-born family members naturalized, or, if any of the foreign-born males registered for the WWI draft, those documents will list the town from which your family came.

3. if you know the name of the cousin that family lore says your grandfather came with, so much the better

Using the above information, head over to Steve Morse's One-Step webpage (https://stevemorse.org/ ).  Steve's search forms will let you search passenger manifests in a variety of different ways.  If the town had a small population, you may be able to locate your grandfather's manifest by using just the year of arrival, his age & the town name, for example.  I've done the same for cases where the name was badly mangled, or if I didn't know the original surname or person's Yiddish name. The Ellis Island form also allows you to search with a companion name.

Remember to search all of the ports -- not everyone landed at Ellis Island!

Regards,
David Oseas

Researching:
HYMAN/HEYMAN/HEIMOWITS/CHAJMOVITS: Zemplen-Dobra, Hungary > New York
KLEIN: Satoraljaujhely (Ujhely), Hungary > New York > Los Angeles
KRONOWITH: Hungary > New York
OSEAS/OSIAS/OSIASI/OZIAS: Iasi, Romania > Chicago > Milwaukee > Los Angeles
SCHECHTER/SHEKTER: Kishinev, Bessarabia > New York  
SHERMAN: Iasi, Romania > New York > Los Angeles
STRUL:  Iasi, Romania > Haifa, Israel
WICHMAN: Syczkowo (Bobruisk), Belarus > Milwaukee > Los Angeles