Search relies on indexing to return relevant records. A limited set of fields and types are indexed for search on each individual object by default and are outlined in the following documentation:
Salesforce Classic - Searchable Fields by Object in Salesforce Classic
Lightning - Search Manager shows you a list of available searchable fields in your org
Since fields of type Text for custom objects are always indexed for search, an administrator may consider creating a custom text field and populating it with a value or data contained in another field (that's not indexed by default) via a flow.
For example, say users want to see Contacts related to an Account by searching Account Name as outlined in article Cannot see all Contacts associated to an Account when I search for Account Name or perhaps users would like to find related Opportunities by searching Account Name or for records containing a particular picklist field's value or other field type that's not typically indexed for search by default.
1. Create a custom text field on the object that you're searching records. Since this field will only be used for searching purposes, don't add the field to your Page Layouts.
2. Create a Flow with the following criteria to update your custom text field with the value that you wish to search for. See Build a Flow for more details.
Lastly, you will need to update the newly created text field for all existing records. See Mass update records in Salesforce and How to retroactively trigger workflow on already existing records? for more details.
If users are still unable to locate desired records see Unable to find records in global search for additional troubleshooting recommendations.
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