If you are awarded the Armed Forces Expeditionary Medal, are you authorized a combat patch?

TL;DR: If the AFEM eligibility criteria does not match the list for SSI-MOHC, then no, the "combat patch" shall not be awarded until such time as the criteria is amended/updated.

The AFEM is authorized for the following operations (Per AR 600-8-22):

Quemoy and Matsu Islands 23 August 1956 to 1 June 1963
Lebanon 1 July to 1 November 1958
Taiwan Straits 23 August 1958 to 1 January 1959
Berlin 14 August 1961 to 1 June 1963
Cuba 24 October 1962 to 1 June 1963
Congo 23 to 27 November 1964
Dominican Republic 28 April 1965 to 21 September 1966
Korea 1 October 1966 to 30 June 1974
Cambodia-Operation EAGLE PULL 11 to 13 April 1975 Evacuation of Cambodia.
Vietnam-Operation FREQUENT WIND 29 to 30 April 1975 Evacuation of Vietnam
Mayaguez Operation 15 May 1975
Grenada-Operation URGENT FURY 23 October to 21 November 1983 The qualifying criteria for non-unit direct support personnel in Grenada are 6 consecutive days or 12 nonconsecutive days.
Libya-Operation ELDORADO CANYON 12 to 17 April 1986
Panama-Operation JUST CAUSE 20 December 1989 to 31 January 1990
Haiti-Operation UPHOLD DEMOCRACY 16 September 1994 to 31 March 1995
Congo 14 July 1960 to 1 September 1962
Somalia-Operations RESTORE HOPE and UNITED SHIELD 5 December 1992 to 31 March 1995
Operation JOINT ENDEAVOR (OJE) 20 November 1995 to 19 December 1996 See note 1. Only for participants deployed in support of the Former Republic of Yugoslavia in Bosnia-Herzegovina.
Exercise Intrinsic Action and Iris Gold:
1 December 1995 to 15 February 1997
31 December 1997 to 15 July 1998
11 November 1998 to 22 December 1998
Operation JOINT GUARD (OJG) 20 December 1996 to 20 June 1998 See note 1. Only for participants deployed in support of the Former Republic of Yugoslavia in Bosnia-Herzegovina.
Operation JOINT FORGE (OJF) 21 June 1998 to 2 December 2004
Vietnam 1 July 1958 to 3 July 1965
Laos 19 April 1961 to 7 October 1962
Cambodia 29 March to 15 August 1973
Thailand 29 March to 15 August 1973 Only those in direct support of Cambodia operations.
El Salvador 1 January 1981 to 1 February 1992
Lebanon 1 June 1983 to 1 December 1987
Persian Gulf-Operation EARNEST WILL 24 July 1987 (the date of the Bridgeton incident) to 1 August 1990 The area of operation is the area from 20 degrees north latitude northward to 30 degrees, 30 minutes, north latitude and from 46 degrees, 36 minutes, east longitude eastward to 63 degrees east longitude. These geographical limits include the Persian Gulf, Bahrain, Kuwait, the Gulf of Oman, and most of Saudi Arabia.
SWA operations:
OSW 1 December 1995 to 18 March 2003
Maritime Intercept Operation (MIO) 1 December 1995 to 18 March 2003
Vigilant Sentinel 1 December 1995 to 15 February 1997
Operation NORTHERN WATCH (ONW) 1 January 1997 to 18 March 2003 The AOE includes the land area within Incirlik Air Base, Turkey; the total land area and airspace of the countries of Iraq, Saudi Arabia and Kuwait, and that portion of the Arabian Gulf (Persian Gulf) that lies west of the 56-degree east longitude.
Operation DESERT THUNDER 11 November to 22 December 1998
Operation DESERT FOX 16 to 22 December 1998
Operation DESERT SPRING 31 December 1998 to 18 March 2003
Operation SECURE TOMORROW 29 February 2004 to 15 June 2004
Operation JOINT GUARDIAN and NATO Headquarters 1 January 2014 to a date to be determined

Per AR 670-1, the SSI-MOHC "Combat Patch" is authorized for the following designated campaigns:
World War II: between 7 December 1941 and 2 September 1946.
Korea: between 27 June 1950 and 27 July 1954. Also between 1 April 1968 and 31 August
1973, for those personnel who were awarded the Purple Heart, combat infantryman badge, combat medical badge, or who qualified for at least one month of hostile fire pay for service in a hostile fire area in Korea.
The Dominican Republic: between 29 April 1965 and 21 September 1966
Grenada, to include the Green and Carriacou Islands: between 24 October 1983 and 21 November 1983
Lebanon: between 6 August 1983 and 24 April 1984
Korea: 23 November 1984, for Soldiers who participated directly in the firefight with North Korean guards at the Joint Security Area, Panmunjom, Korea
Persian Gulf: between 27 July 1987 and 1 August 1990
Panama: between 20 December 1989 and 31 January 1990
The Persian Gulf: between 17 January 1991 and 31 August 1993
El Salvador: between 1 January 1981 and 1 February 1992
Somalia: from 1 January 2004 to a date to be determined (there is more info listed in the regs for this)
Operation Enduring Freedom (OEF): between 11 September 2001 and 31 December 2014
Operation Iraqi Freedom (OIF): between 19 March 2003 and 31 August 2010
Operation New Dawn (OND): between 1 September 2010 and 31 December 2011
Operation Inherent Resolve: between 15 June 2014 and a date to be determined
Operation Freedom’s Sentinel (OFS): between 1 January 2015 and a date to be determined.

The award of AFEM does not by itself warrant authorization to wear a shoulder sleeve insignia–former wartime service (SSI–FWTS). AFEM indicates participation in certain military expeditions but it doesn't necessarily signify service in a designated combat zone that would grant eligibility for a combat patch. Refer the Soldier to AR 670-1 para 19-17 for SSI–FWTS eligibility.

Wouldn't it be nice if it was that clear cut?

I don't think it is though.