I had a unlawful detainer notice at a apartment. Will a new landlord rent to me or will they all say no?

It realy depends upon the specific landlord. Just because you had a recent eviction on your record does not necessarily mean that all landlords will refuse to rent to you. Perhaps you have a good explanation for what happened? As you suggest, offering a higher security deposit (maximum is 2 months' rent for an unfurnished unit) and/or offering to prepay one or more months' rent might convince an otherwise wary landlord to rent to you.

As Frank said, it really depends on the landlord. Most landlords use your rental history, credit score, and proof of income as variable to determine if you will be a good renter but there is nothing legally speaking from preventing them from using their discretion, other than a corporate policy, to rent to someone who may have a less than perfect background based on the application. An explanation of why you were evicted in the first place can sometimes mitigate those fears along with an increased security deposit. Hope it works out!

As Attorney Chen suggested, make sure that if a landlord/management company is willing to rent to you that they do not take unfair or unlawful advantage of you knowing you are possibly desperate your search for an apartment/housing. Make sure they stay within bounds with respect to the security deposit (as Chen mentioned) and if you are ever suspicious that you are being offered terms of a lease that may be unlawful, speak to an attorney immediately.