code.rb

Error = Class.new(StandardError)

class Client
  Error = Class.new(StandardError)

  def call
    raise "ZOMG"
  rescue => e
    raise Error.new("Uh oh #{e.inspect}")
  end
end

begin
  Client.new.call
rescue Client::Error => e
  raise ::Error.new("Uh oh #{e.inspect}")
end

output.txt

tmp/raise-cause.rb:16:in `rescue in <main>': Uh oh #<Client::Error: Uh oh #<RuntimeError: ZOMG>> (Error)
	from tmp/raise-cause.rb:13:in `<main>'
tmp/raise-cause.rb:9:in `rescue in call': Uh oh #<RuntimeError: ZOMG> (Client::Error)
	from tmp/raise-cause.rb:6:in `call'
	from tmp/raise-cause.rb:14:in `<main>'
tmp/raise-cause.rb:7:in `call': ZOMG (RuntimeError)
	from tmp/raise-cause.rb:14:in `<main>'