Using tab for both cycling and completion in org mode

Question

I use company for completion and I have company-indent-or-complete-common bound to tab for certain modes. In org mode, tab is bound to org-cycle, which either cycles visibility or calls the global binding for tab depending on context. I would like to have org-cycle call company-indent-or-complete-common instead of the global binding for tab, but I don't know how to do this. Is there a simple way to trick org-cycle into having this behavior?

Edit: I managed to find a solution on my own (for some reason @tjg's answer doesn't work for me). At first I was using

(defun org-cycle-advice (old-fn &rest r)
  (let ((old-tab (global-key-binding (kbd "TAB"))))
    (global-set-key (kbd "TAB") 'company-indent-or-complete-common)
    (apply old-fn r)
    (global-set-key (kbd "TAB") old-tab)))

(advice-add 'org-cycle :around 'org-cycle-advice)

but that didn't generalize to outshine-cycle well, so now I am using

(defun company-outline-cycle-indent-or-complete-common ()
  (interactive)
  (if (not (outline-on-heading-p))
      (if (string= major-mode "org-mode")
          (org-cycle)
        (if (bound-and-true-p outshine-mode)
        (outshine-cycle)))))

Answer 1

My approach to this has been to define a minor-mode to put expansion on tabs, which Org will then use as the "default" if TAB is called somewhere not cyclable.

(defvar org-expand-on-tab-minor-mode-map
  (let ((map (make-sparse-keymap)))
    (define-key map (kbd "TAB") 'company-indent-or-complete-common)
      map)
  "Redefine the tab key locally")

(define-minor-mode org-expand-on-tab-minor-mode
  "Minor mode to put expansion on TAB"
  :init-value nil
  :lighter " expand-on-tab")

(add-hook 'org-mode-hook '(lambda () (org-expand-on-tab-minor-mode t)))