my_awesome_middleware.rb

class MyAwesomeMiddleware
  def initialize(app)
    @app = app
  end

  def call(env)
    return_value = @app.call(env)

    do_awesome_post_request_processing

    return_value
  end

  def do_awesome_post_request_processing
    # ...
  end
end

rackup.ru

require 'my_awesome_middleware'

require 'sinatra'
require 'my_sinatra_app'

use MyAwesomeMiddleware

run Sinatra::Application