If the domain name is not with any auction house partner (I think that’s what you’re saying), then you can use a drop catch service anytime: https://www.domainsherpa.com/domain-name-backorder-services/ They do not make your order public information as NameJet and others do by sharing how many bids are placed on a domain name and thereby driving interest.
The easiest and least time consuming option is to 301 redirect the old domain to your existing site. This tactic obviously works best if both sites are in the same sector and are targeting the same keywords; otherwise, if you have a pet supply site and you buy an old Texas Hold 'Em poker site, a redirect probably might raise some eyebrows among the search engines. If, however, your site is brandnamepets.com and you buy onlinepetsupply.com and 301 redirect the domain over, you're inheriting a lot of topical and appropriate links. 

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